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Problems in classical potential theory with applications to mathematical physics
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Lundberg, Erik
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Cauchy Problem
Dirichlet Problem
Fischer Operator
Gravitational Lensing
Laplacian Growth
Quadrature Domain
Dissertations, Academic -- Mathematics Physics Astrophysics -- Doctoral -- USF   ( lcsh )
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ABSTRACT: In this thesis we are interested in some problems regarding harmonic functions. The topics are divided into three chapters. Chapter 2 concerns singularities developed by solutions of the Cauchy problem for a holomorphic elliptic equation, especially Laplace's equation. The principal motivation is to locate the singularities of the Schwarz potential. The results have direct applications to Laplacian growth (or the Hele-Shaw problem). Chapter 3 concerns the Dirichlet problem when the boundary is an algebraic set and the data is a polynomial or a real-analytic function. We pursue some questions related to the Khavinson-Shapiro conjecture. A main topic of interest is analytic continuability of the solution outside its natural domain. Chapter 4 concerns certain complex-valued harmonic functions and their zeros. The special cases we consider apply directly in astrophysics to the study of multiple-image gravitational lenses.
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Disseration (Ph.D.)--University of South Florida, 2011.
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Includes bibliographical references.
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by Erik Lundberg.
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Problems in classical potential theory with applications to mathematical physics
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Includes bibliographical references.
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ABSTRACT: In this thesis we are interested in some problems regarding harmonic functions. The topics are divided into three chapters. Chapter 2 concerns singularities developed by solutions of the Cauchy problem for a holomorphic elliptic equation, especially Laplace's equation. The principal motivation is to locate the singularities of the Schwarz potential. The results have direct applications to Laplacian growth (or the Hele-Shaw problem). Chapter 3 concerns the Dirichlet problem when the boundary is an algebraic set and the data is a polynomial or a real-analytic function. We pursue some questions related to the Khavinson-Shapiro conjecture. A main topic of interest is analytic continuability of the solution outside its natural domain. Chapter 4 concerns certain complex-valued harmonic functions and their zeros. The special cases we consider apply directly in astrophysics to the study of multiple-image gravitational lenses.
538
Mode of access: World Wide Web.
System requirements: World Wide Web browser and PDF reader.
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Advisor:
Khavinson, Dmitry .
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Dirichlet Problem
Fischer Operator
Gravitational Lensing
Laplacian Growth
Quadrature Domain
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ProblemsinClassicalPotentialTheory withApplicationsto MathematicalPhysics by ErikLundberg Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofMathematics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:DmitryKhavinson,Ph.D. CatherineBeneteau,Ph.D. VilmosTotik,Ph.D. EvgueniiRakhmanov,Ph.D. DateofApproval: November16,2010 Keywords:Quadraturedomain,Schwarzpotential,Laplaciangrowth,Zerner's Theorem,Bony-ShapiraTheorem,globalizingfamily,Fischeroperator,Gauss decomposition,Almansidecomposition,Brelot-ChoquetTheorem,polyharmonic function,lightningbolt,harmonicmap,Blaschkeproduct,gravitationallensing c Copyright2011,ErikLundberg

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Acknowledgements Wordscan'texpressmygratitudetoDmitryKhavinsonforbeingmyadvisor. Iwasinitiallydrawntohisshowmanshipinfrontofanaudienceandhislarger-thanlifepersonality;havingspentthepastfouryearsashisstudent,Iamamazedthathe seemsnevertorunoutoftricks,bothasamathematicianandasanentertainer.I knowthatIamluckytohavehadanadvisorwiththeenergyofyouthandwisdom ofage.Atthesametime,hecanbekind,concerned,andgenerous.Hisguidancehas encouragedmeimmensely. IwouldalsoliketothanktheotherfacultyattheUniversityofSouthFlorida whohavebeensupportiveandsurprisinglyhospitablewheneverIinvadedtheirofcehoursonshortnotice.Inadditiontothemembersofmycommittee,Catherine Beneteau,EvgueniiRakhmanov,andVilmosTotik,IamgratefultoThomasBieske, BrendanNagle,SherwinKouchekian,RazvanTeodorescu,andDavidRabson. IwishtoacknowledgetheinspirationIhavereceivedfrommycorrespondence withHermannRenderandtheimpactithadonthethirdchapter.Discussionswith MarkMineevprovidedmewithanexcitingintroductiontothephysicsofLaplacian Growthwhichledtothesecondchapter.Inregardstochapterfour,Iwouldlike tothankWalterBergweiler,AlexEremenko,andCharlesR.Keetonforstimulating discussionsandforlettingmeseesomeoftheirunpublishedwork.

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TableofContents Abstractiv 1Introduction1 1.1SingularitiesoftheSchwarzpotentialandLaplaciangrowth.....4 1.2AlgebraicDirichletProblems.......................7 1.3ValenceofHarmonicMapsandGravitationalLensing.........10 2SingularitiesoftheSchwarzpotentialandLaplacianGrowth14 2.1LaplacianGrowth.............................14 2.2DynamicsofSingularities.........................17 2.2.1TheSchwarzPotential......................17 2.2.2LaplaciangrowthandtheSchwarzpotential..........18 2.2.3ACauchyproblemconnectedtoEllipticGrowth........21 2.3Examples.................................24 2.3.1Laplaciangrowthintwodimensions...............24 2.3.2Examplesandnon-examplesin R 4 ................26 2.3.3Examplesofellipticgrowth....................30 2.4TheSchwarzpotentialin C n .......................32 2.5Quadraturedomains...........................38 2.6Concludingremarks............................41 i

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3AlgebraicDirichletProblems45 3.1TheSearchforSingularitiesofSolutionstotheDirichletProblem:RecentDevelopments............................45 3.1.1Themainquestion.........................45 3.1.2TheCauchyProblem.......................46 3.1.3TheDirichletproblem:Whendoesentiredataimplyentire solution?..............................47 3.1.4Whendoespolynomialdataimplypolynomialsolution?....48 3.1.5Dirichlet'sProblemandOrthogonalPolynomials........49 3.1.6LookingforsingularitiesofthesolutionstotheDirichletProblem52 3.1.7Render'sbreakthrough......................54 3.1.8Backto R 2 :lightningbolts....................56 3.1.9Furtherquestions.........................60 3.2Dirichlet'sProblemandComplexLightningBolts...........60 3.2.1AlgebraicallyposedDirichletproblems.............61 3.2.2RealLightningBoltsandIll-posedProblemsfortheWaveEquation61 3.2.3ComplexLightningBoltsandtheVekuahull..........65 3.2.4Ebenfelt'sExampleRevisited...................66 3.2.5AFamilyofCurvesNotCoveredbyRender'sTheorem....69 3.3TheKhavinson-ShapiroConjectureandPolynomialDecompositions.75 3.3.1AlgebraicDirichletproblemsandPolyharmonicDecompositions75 3.3.2Fischeroperatorsandharmonicdivisors.............78 3.3.3Proofofthemainresult......................81 3.3.4Criteriafordegree-relateddecompositions............83 ii

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4ValenceofHarmonicMapsandGravitationalLensing89 4.1FixedPointsofConjugatedBlaschkeProductswithApplicationsto GravitationalLensing...........................89 4.1.1Case r z = B z .........................91 4.1.2GravitationalLensingbyCollinearPointMasses........94 4.2TranscendentalHarmonicMappingsandGravitationalLensingbyIsothermalGalaxies................................97 4.2.1Asimpleproblemincomplexanalysiswithadirectapplication97 4.2.2Preliminaries:TheArgumentPrinciple.............99 4.2.3AnUpperBoundfortheNumberofImages...........100 4.2.4Remarks..............................104 4.2.5Derivationofthecomplexlensingequationfortheisothermal ellipticalgalaxy..........................105 References108 iii

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Abstract Inthisthesisweareinterestedinsomeproblemsregardingharmonicfunctions. Thetopicsaredividedintothreechapters. Chapter2concernssingularitiesdevelopedbysolutionsoftheCauchyproblemforaholomorphicellipticequation,especiallyLaplace'sequation.Theprincipal motivationistolocatethesingularitiesoftheSchwarzpotential.Theresultshave directapplicationstoLaplaciangrowthortheHele-Shawproblem. Chapter3concernstheDirichletproblemwhentheboundaryisanalgebraicset andthedataisapolynomialorareal-analyticfunction.Wepursuesomequestions relatedtotheKhavinson-Shapiroconjecture.Amaintopicofinterestisanalytic continuabilityofthesolutionoutsideitsnaturaldomain. Chapter4concernscertaincomplex-valuedharmonicfunctionsandtheirzeros. Thespecialcasesweconsiderapplydirectlyinastrophysicstothestudyofmultipleimagegravitationallenses. iv

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1Introduction Thisthesisinvestigatesafewproblemsinpotentialtheoryandcomplexanalysis. ThequestionsspanCauchyandDirichletproblemsforLaplace'sequation,realand complex-valuedharmonicfunctions,singularitiesandzeros,andweconsiderboththe two-dimensionalcaseandarbitrarydimensions.Wearenotsomuchinterestedin pathologiesorinseekinggeneralityforitsownsake.Ratherthegoalhasbeentogain someinsightonafewproblemsthatarephysicallymotivatedandsimpletostate.We dividethetopicsintothreechapters. Chapter2: concernssingularitiesofCauchy'sproblemfortheLaplaceequationand isbasedonthepaper[78],whichhasbeensubmittedforpublication. Chapter3: concernssingularitiesandalgebraicityofDirichletproblemsandconsists ofthepapers[57],[77],and[79]. Chapter4: concernscomplex-valuedharmonicfunctionsarisinginmodelsofgravitationallensingandconsistsofthepapers[71]and[58]. Althoughthetopicsbetweenchaptersaresomewhatseparated,acommon threadisaninteractionbetweenalgebraicgeometryandharmonicfunctions;intwodimensionalinstancesofsuchsituations,theSchwarzfunctioncanoftenplayarole. Therefore,beforegivingamoredetailedoverviewofeachchapter,letushaveabrief glimpseoftherangeoftopicsbydeningtheSchwarzfunctionandseeinghowitcan ariseineachcontext. Suppose)-290(isareal-analyticcurveintheplane.ThenP.Davis[24]hasdened the Schwarzfunction of)-415(tobetheuniquefunction,complex-analyticinaneighborhoodof,thatcoincideswith z on,where z denotesthecomplexconjugateof 1

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z .If,forinstance,)-398(isgivenalgebraicallyasthezerosetofapolynomial P x;y wecanobtain S z bymakingthecomplex-linearchangeofvariables z = x + iy z = x )]TJ/F27 11.9552 Tf 11.808 0 Td [(iy ,andthensolvingfor z intheequation P z + z 2 ; z )]TJ/F25 7.9701 Tf 7.077 0 Td [( z 2 i =0.Letusconsidera fewexamples. Example1: Suppose)-449(isthecurvegivenalgebraicallybysolutionsetof x 2 + y 2 2 = a 2 x 2 + y 2 +4 2 x 2 C.Neumann'soval".Thenforappropriatevalues of a and ,)-345(isasingleclosed,boundedcurve.Changingvariableswehave z z 2 = a 2 z z + 2 z + z 2 .Solvingfor z gives S z = z a 2 +2 2 + z p 4 a 4 +4 a 2 2 +4 2 z 2 2 z 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(" 2 S z has twosimplepolesintheinteriorof)-326(andasquarerootbranchcutintheexterior. Example2: Suppose)-320(isanellipsegivenbythesolutionsetoftheequation x 2 a 2 + y 2 b 2 =1.Thenchangingvariableswehave z + z 2 a 2 )]TJ/F25 7.9701 Tf 13.047 5.698 Td [( z )]TJ/F25 7.9701 Tf 7.078 0 Td [( z 2 b 2 =4.Solvingfor z gives S z = a 2 + b 2 a 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(b 2 z + 2 ab b 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(a 2 p z 2 + b 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(a 2 .Thus S z hasasquarerootbranchcutalong thesegmentjoiningthefoci p a 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(b 2 Example3: Suppose)-390(istheimageoftheunitcircleundertheconformal map f = a + b 2 ,with a and b bothreal, a> 2 b> 0.Then)-343(isasingleclosed, boundedcurve.In -planecoordinates,theSchwarzfunctionissimply f 1 = a + b 2 Indeedthecondition S z j )]TJ/F19 11.9552 Tf 9.107 1.794 Td [(= z canbepulledback: S f j j j =1 = f ,andonthe unitcircle f coincideswith f 1 since a and b arereal. InChapter2wediscussaprinciplethatreducesthe Laplaciangrowth or HeleShawmovingboundaryproblem toasimpledynamicaldescriptionofthesingularities oftheSchwarzfunction.Thisallowsthegenerationofexplicitexactsolutionsof Laplaciangrowth.Forinstance,basedonthecalculationsabove,onecanselectoneparameterfamiliesofeachoftheExamples1-3toobtainexactsolutions.InExample 1,xing andletting a decrease,theSchwarzfunctionhastwosimplepolesinside )-469(withxedpositionsanddecreasingresidues.Aswewillsee,thisallowsusto interpretatethisoneparameterfamilyasthemovingboundaryofashrinkingdomain ofoilsurroundedbywaterwithsuctionoccurringateachofthesinks"positioned at z = ThemaingoalofChapter2istostudyhigher-dimensionalLaplaciangrowth 2

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intermsofageneralizationoftheSchwarzfunction. InChapter3weareinterested,inparticular,intheclassicalDirichletproblem posedonanalgebraiccurvewithpolynomialdata,andthequestionisifandwhere thesolutiondevelopssingularities.IfwemultiplytheSchwarzfunctioninExample1 by z 2 )]TJ/F27 11.9552 Tf 12.123 0 Td [(" 2 ,thenweobtainafunction f z = z 2 )]TJ/F27 11.9552 Tf 12.123 0 Td [(" 2 S z analyticintheinterior of)-352(andcoincidingon)-353(with p z; z = z 2 )]TJ/F27 11.9552 Tf 12.169 0 Td [(" 2 z ,apolynomial.Thusthesolution totheDirichletproblemwithpolynomialdata p developssingularitiesatthebranch cutsof S z .Ifwewantreal-valueddataandsolution,thenwecantakethereal partof f and p Thisgivesanillustrativeexample,butinChapter3wewillbeconsidering Dirichlet'sproblemposedonclassesofcurvesandsurfacesforwhichthistrickis unavailable. InChapter4,inordertocalculatethegravitationallensingeectofamassive object,abasicproblemthatarisesistocalculatethe Cauchytransform ofatwodimensionaldomain. C = 1 Z dA z )]TJ/F27 11.9552 Tf 11.955 0 Td [(z Suppose 2 c andapplyGreen'sTheorem: 1 Z dA z )]TJ/F27 11.9552 Tf 11.955 0 Td [(z = 1 2 i Z @ zdz )]TJ/F27 11.9552 Tf 11.955 0 Td [(z IftheboundaryofhasaSchwarzfunction,thenwecanreplace z with S z .ThentheCauchytransformcanbedeterminedfromthesingularitiesofthe Schwarzfunctionin.ForExample1above,wehave C = Const 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(" + 1 + .In thecontextofChapter4thismeansthatthegravitationallensconsistingofanobject withuniformmasssupportedoverproducesthesamelensingeectoutsideits supportasdotwopointmasses.InChapter4,wewillbemoreinterested,though,in theCauchytransformofacertain non-uniform massdensitysupportedonanellipse. TheSchwarzfunctionalsoplaysaroleinthiscase. 3

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1.1SingularitiesoftheSchwarzpotentialandLaplaciangrowth TheSchwarzfunctioncanbepartiallygeneralizedtohigherdimensionsusingthe followingCauchyproblemposedinthevicinityof,whereweassume)-460(isnonsingularandanalytic.ThesolutionexistsandisuniquebytheCauchy-Kovalevskaya Theorem. 8 > > > < > > > : w =0near)]TJ/F27 11.9552 Tf -21.766 -20.922 Td [(w j )]TJ/F19 11.9552 Tf 9.107 1.793 Td [(= 1 2 jj x jj 2 r w j )]TJ/F19 11.9552 Tf 9.107 1.793 Td [(= x .1.1 where= P n j =1 @ 2 @x 2 j istheLaplacian. Denition1.1.1 Thesolution w x oftheCauchyproblem1.1.1iscalledthe Schwarz Potential of. In R 2 ,theSchwarzfunctioncanbedirectlyrecoveredfromtheSchwarzpotential.Consider S z =2 @ z w = w x )]TJ/F27 11.9552 Tf 11.494 0 Td [(iw y .TheCauchy-Riemannequationsfor S follow fromharmonicityof w ,and r w = x on)-327(implies S z = z on. Example: Let)-278(:= f x 2 R n : jj x jj 2 = r 2 g beasphereofradius r .When n = 2,itiseasytoverifythat w z = r 2 log j z j +1 = 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(log r solvestheCauchyProblem .1.1,andinhigherdimensionstheSchwarzpotentialis w x = )]TJ/F28 7.9701 Tf 31.724 4.707 Td [(r n n )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 jj x jj n )]TJ/F26 5.9776 Tf 5.756 0 Td [(2 + n 2 n )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 r 2 ThisgivesapartialgeneralizationoftheSchwarzfunction.ThereectionprincipleassociatedwiththeSchwarzfunctiondoesnotgeneralizetohigherdimensions bythisoranyothermeans,buttheSchwarzpotentialretainsotherpropertieswe areinterestedin.Forinstance,theSchwarzfunctionplaysanimportantroleinthe Laplaciangrowthproblemintheplane.TobebriefseeChapter2fordetails,inthe LaplacianGrowthLGproblem,thevelocityofamovingboundaryisdeterminedat eachinstantbythegradientofitsGreen'sfunctionpressure"withxedsingularity. Thetime-derivativeoftheSchwarzfunctionofthemovingboundarycoincideswith 4

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the z -derivativeofthepressure,leadingtoareformulationofLaplaciangrowthin termsofasimpledescriptionofthetime-dependenceofsingularitiesoftheSchwarz function.Thisprovidesauniedviewofthemanyclassicalexactsolutionsandhas leadtofurthersystematicdevelopments. Inhigherdimensions,LGcontinuestobegovernedbythesingularitiesofthe Schwarzpotential.In[78],wegaveasimpleproofofthefollowing. Theorem1.1.2 If w x ;t istheSchwarzpotentialof @ t then t solvestheLaplacian growthproblemifandonlyif @ @t w x ;t = )]TJ/F27 11.9552 Tf 9.298 0 Td [(nP x ;t .1.2 where n isthespatialdimension.Inparticular,singularitiesoftheSchwarzpotential inthe t donotdependontime,exceptforonestationedatthesource/sinkwhich doesnotmovebutsimplychangesstrength. In[78]westudythesingularitiesoftheSchwarzpotentialwiththegoalof obtainingapplicationstoLGusingTheorem1.1.2.Indimensionshigherthantwo, thesingularitysetoftheSchwarzpotentialismuchmoremysteriousandthereisa deciencyofexactsolutionstoLG,whereasintwodimensionsthereisanabundance ofexplicitexamples. FollowingL.Karp,weconsiderthespecialcaseofaxially-symmetricexamples in R 4 ,whichcanbereducedtothesingularitiesoftheSchwarzfunctionofthecurve thatgeneratesthehypersurfaceofrevolution.WedescribesomeexamplesofLGin R 4 Forthree-dimensionalexamples,onlyquadraticsurfaceshavebeenunderstood. G.Johnsson[52]gavethecompletedescriptionforquadraticsurfacesbyglobalizing theproofofLeray'sprincipleinthiscase.Inordertostudysomesurfacesofhigher degree,in[78]welifttheproblemto C n andusetheglobalizingtechniqueofBony andSchapira[19]combinedwiththelocalextensionTheoremofZerner[107]andthe 5

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morerecentTheoremofEbenfelt,Khavinson,andShapiro[31].Weareabletoprove thefollowingregardingsurfacesofrevolutiongeneratedbyC.Neumann'soval". Theorem1.1.3 Let W x betheSchwarzpotentialoftheboundary )]TJ/F44 11.9552 Tf 11.246 0 Td [(ofthedomain := f x 2 R n : P n i =1 x 2 i 2 )]TJ/F27 11.9552 Tf 12.644 0 Td [(a 2 P n i =1 x 2 i )]TJ/F19 11.9552 Tf 12.644 0 Td [(4 x 2 1 < 0 g .Then W canbeanalytically continuedthroughout n B where B isthesegment f x 1 2 [ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1] ;x j =0 for j = 2 ;::;n g In[78]wealsoconsiderageneralizationofLGtothecasewhenthephysical propertiesofthemediumarenon-homogeneous.Thisistheso-called ellipticgrowth problem.WeareabletogeneralizeTheorem1.1.2tothiscaseandobtainsomenovel explicitexactsolutions. Twoconjecturesinpotentialtheoryariseinourstudy.Therstconjecture suggestsaresponsetoH.S.Shapiro'sremarkthatitisnotknownwhetherquadraturedomainsdomainswhoseSchwarzpotentialhasnitelymanynite-orderpointsingularitiesinthedomainarealwaysalgebraicindimensionshigherthantwo. Conjecture1.1.4 Indimensionsgreaterthantwo,thereexistquadraturedomains thatarenotalgebraic. ThefollowingconjecturegeneralizestheSchwarzPotentialConjectureformulatedbyKhavinsonandShapiro.Iftrue,itwouldnaturallyseparateellipticgrowth problemsintoclassesseeChapter2fordetails.TheSchwarzpotentialConjecture iseasytoproveintheplane,butwedonotknowifthefollowingistrueevenintwo dimensions. Conjecture1.1.5 Suppose > 0 isentireandthat u solvestheCauchyproblemona nonsingularanalyticsurfacefordiv r u =0 withentiredata.Thenthesingularity setof u iscontainedinthesingularitysetof v ,where v solvestheCauchyproblem withdatafunction q ,whichisaglobalsolutionofdiv r q =1 6

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1.2AlgebraicDirichletProblems Letbeasmoothlyboundeddomainin R n .ConsidertheDirichletProblemDP inofndingthefunction u ,say, 2 C 2 T C andsatisfying 8 < : u =0 u j )]TJ/F19 11.9552 Tf 9.108 1.793 Td [(= v ; .2.3 where= P n j =1 @ 2 @x 2 j and)-383(:= @ ;v 2 C \051.Thesolution u existsandisunique, andif)-416(:= @ isacomponentofareal-analytichypersurfaceandthedata v is real-analyticinaneighborhoodof ,then u extendsasareal-analyticfunctionacross @ intoanopenneighborhood 0 of ItisclassicallyknownthattheDirichletproblemforLaplace'sequationwith polynomialdataposedonanellipsoidhasapolynomialsolution.D.Khavinsonand H.S.Shapiroshowedthatreal-entiredatagiveareal-entiresolutionforellipsoids[64]. Theyformulatedaconjecturethat,intermsoftheDirichletproblem,ellipsoidsare characterizedwithintheclassofboundeddomainsbyeachofthepropertiesithat anypolynomialdatahasapolynomialsolutionandiianyentiredatahasanentire solution. Chapter3consistsofthepapers[57],[77],and[79].In[57]writtenjointly withDmitryKhavinson,wegiveadetailedsurveyofworkrelatedtoconjectures iandii.Thepaper[79]jointworkwithHermannRenderrelatestoconjecture i,andthepaper[77]relatestoconjectureii.Alongthelinesoftheconjecture ii,itisnaturaltoaskthefollowingquestiononacasebycasebasisfordierent algebraically-boundeddomains: Question: Givenadomainwithalgebraicboundaryandpolynomialorentiredata, doesthesolutiontotheDirichletproblemdevelopsingularities,andifso,howfar outsidetheinitialdomainarethey? Thisquestionsetsthetoneforthepaper[77],whereweconsiderspecicexamplesintheplaneandanswerthequestionbyspecifyingdataandestimatingthe 7

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positionofthesingularitiesthatdevelop.Theestimatesuse C 2 -techniquesinvolvingannihilatingmeasuressupportedonspecialnitesetsofpointscalledlightning bolts".ThisisacomplexversionofthelightningboltsintroducedbyKolmogorov andArnoldtosolveHilbert's13thproblem.Thecomplexlightningboltswererst usedbyHansenandShapiro[46]toshowfailureofanalyticcontinuabilityofharmonic functionsbutwithoutestimatingthelocationofsingularities.Inordertousecomplexlightningboltstoestimatethelocationofsingularities,in[77]wealsostudythe geometryofthecomplexicationofthecurvein C 2 inrelationshiptotherealsection ofthecurveandtheVekuahullofcertaindomainsin R 2 .Thisissummarizedinthe prescriptionprovidedbythefollowingTheorem,where ~ )-295(isthe complexication of, i.e.thezerosetin C 2 ofthesamepolynomial,and ^ isthe VekuaHull of. Theorem1.2.1 Let )]TJ/F25 7.9701 Tf 7.314 -1.793 Td [(1 beaconnectedcomponentofthealgebraiccurve )]TJ/F44 11.9552 Tf 7.314 0 Td [(,andlet beasimplyconnecteddomain.Suppose ~ )]TJ/F44 11.9552 Tf 10.845 0 Td [(containsaclosedlightningboltwithrespect tothecomplex-characteristiccoordinates z and w oflength 2 n .Supposefurtherthat along ~ )]TJ/F44 11.9552 Tf 12.036 0 Td [(therearepaths,alsocontainedin ^ ,thatconnecteachvertexto )]TJ/F25 7.9701 Tf 7.314 -1.794 Td [(1 .Then, fortheDirichletproblemon )]TJ/F25 7.9701 Tf 7.314 -1.793 Td [(1 ,thereexistpolynomialdataofdegree n whosesolution cannotbeanalyticallycontinuedtoallof UsingthisTheorem,weestimatethelocationofsingularitiesdevelopedbythe analyticcontinuationofsolutionswithcertaindataposedonafewdierentfamilies ofcurves. Example1: )-452(isthesolutionsetof p x 4 + q y 4 =1where p x and q x are positivefor x 2 R + andsatisfy p + q = p + q =1.FortheDirichlet problemwithnon-harmonic,quadraticdataonthecurve p x 4 + q y 4 =1,the solutiondevelopssingularitiesonthe x and y axesnofurtherfromtheoriginthan max fj x j + j y j : p x 4 + q y 4 =1 g .AspecialcaseistheTV-screen" x 4 + y 4 =1 consideredbyP.Ebenfeltin[30],wherethesingularitiesweredescribedexhaustively. Example2: )-324(isthezerosetof P x;y = x 2 + y 2 )]TJ/F19 11.9552 Tf 11.9 0 Td [(1 )]TJ/F19 11.9552 Tf 11.9 0 Td [(2 x 3 + xy 2 )]TJ/F27 11.9552 Tf 11.9 0 Td [(x 2 + y 2 .For theDirichletproblemwithanynon-harmonic,quadraticdataposedonthebounded 8

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componentof)-467( small,thesolutiondevelopssingularitiesnofurtherfromthe originthan 1+ p 1+2 p 2 )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 ,whichis,asymptotically,twicethedistancefromthebounded componenttotheunboundedcomponent. Example3: )-266(isthesolutionset8 x x 2 )]TJ/F27 11.9552 Tf 10.477 0 Td [(y 2 +57 x 2 +77 y 2 =49aperturbedellipse. FortheDirichletproblemposedinsidetheboundedcomponentof,thereexistcubic dataforwhichthesolutiondevelopsasingularityonthex-axisnofurtherfromthe originthan7 : 622comparetothex-intercept, )]TJ/F19 11.9552 Tf 9.298 0 Td [(7 ; 0,ofthenearestunbounded component.Here,thepolynomialdening)-340(satisesthenecessaryconditionposed byChamberlandandSiegel[22],andthecubicdata y x 2 +77 y 2 )]TJ/F19 11.9552 Tf 12.937 0 Td [(49hasthe polynomialsolution8 xy x 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(y 2 ArecentresultofD.KhavinsonandN.Stylianopoulosprovestheconjecture iintwodimensionsunderanadditionalassumptiononthedegreeofthesolution intermsofthedegreeofthedata.Theirresultgivesaninterestingconsequencefor Fischerdecompositions .Ifapolynomial f hasapolynomialdecomposition f = q + r with r harmonic,then u = f )]TJ/F27 11.9552 Tf 9.493 0 Td [(q solvestheDirichletproblemwithdata f posedonthe zerosetof .Decomposing f as q + r resemblestheinductivestepintheEuclidean algorithm,exceptinsteadofthedegreeconditiondeg r< deg f ,therequirementis that r isharmonic.Inthiscontext,theresultof[65]impliesinparticularthat: FD Givenapolynomial 2 R [ x;y ],ifeverypolynomial f hasaFischerdecomposition f = q + r ,withtheaddedassumptionthatdeg r< deg f + C ,forsome C> 0 independentof f ,thendeg 2. Theproofin[65]usedareformulationintermsofso-callednite-term"recurrencerelationsforBergmanorthogonalpolynomialsandappliedratio-asymptotics involvingtheconformalmaptotheexteriorofthedisk.Thus,theproofreliesheavily onideasfromcomplexanalysis.In[79]jointworkwithHermannRender,weproved thefollowingversionofFDinarbitrarydimensionsandincludinga polyharmonic caseinvolvingthe k thiterateoftheLaplaceoperator: Theorem1.2.2 Let 2 R [ x 1 ;:::;x n ] beapolynomial.Supposethatthereexistsa constant C> 0 suchthatforanypolynomial f 2 R [ x 1 ;:::;x n ] thereexistsadecompo9

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sition f = q f + h f with k h f =0 and deg q f deg f + C: .2.4 Then deg 2 k Theproofusestheassociated Fischeroperator andapplieslinearalgebraand dimensionargumentsinvolvingharmonicdivisors.Wealsoshowedforcertainclasses ofexamplesthatthedegreeconditioninTheorem1.2.2issatised. Theorem1.2.3 Supposethat isapolynomialofdegree t> 2 and = t + s + s )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 + ::: + 0 isthedecompositionintoasumofhomogeneouspolynomials.Assume thepolynomial s isnon-zeroandcontainsanon-negative,non-constantfactor.Let f beapolynomialandassumethatthereexistsadecomposition f = q + h where h isharmonicand q isapolynomial.Then deg q 2 )]TJ/F27 11.9552 Tf 11.986 0 Td [(s +deg f and deg h t +2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(s +deg f: H.Rendersettledbothconjecturesiandiiinarbitrarydimensionsfor thelargeclassofso-called elliptic surfacessurfaceswhosedeningpolynomialhas anonnegativeleadinghomogeneousterm[86].H.S.ShapiroraisedasimplenonellipticexamplenotsettledbyRender'sresults:acircleorsphereperturbedbyany higher-degreehomogeneousharmonicpolynomial.CombiningTheorems1.2.2and 1.2.3showsthat,forthisexample,thereexistspolynomialdataforwhichthesolution isnotapolynomial,conrmingconjectureiinthiscase. 1.3ValenceofHarmonicMapsandGravitationalLensing ThestrongesttestpassedbyEinstein'stheoryofgravitationwasthecorrectprediction ofthedeectionofstarlightasitpassesbyamassiveobject.Besidesbendingor 10

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distortingbackgroundsources,amassiveobjectactingasagravitationallenscan createmultipleimagesofasinglesource.Inmodelingagravitationallens,theeectof movingsomeofthemassparalleltothelineofsightisanorderofmagnitudesmaller thantheeectofmovingitorthogonaltothelineofsight.Thus,evenextremely non-planarsolidmassdensitiessuchasasphericalgalaxycanbeprojectedontoa LensingPlane orthogonaltothelineofsight.Thisisreferredtoasthethin-lens approximation".IntegratingEinstein'sdeectionangleagainsttheprojectedmass densityleadstoa lensingmap sendingtheLensingPlanetotheso-called Source Plane .Lensedimagescanthenbeidentiedaspre-imagesofthesourceposition underthelensingmap. MainProblem: Givenafamilyofgravitationallenses,determinethemaximum numberofimagesthatcanbelensed. Forlightrayspassingoutsidethesupportofthemassdensity,thelensingmap isharmonic.Asasteptowardextendingthefundamentaltheoremofalgebra,D. KhavinsonandG.Neumann[60]usedthetheoryofplanarharmonicmaps combinedwithcomplexdynamicstoproveaboundof5 n )]TJ/F19 11.9552 Tf 11.402 0 Td [(5forthenumberofzeros ofafunctionoftheform r z )]TJ/F19 11.9552 Tf 12.091 0 Td [( z ,where r z isrationalofdegree n .Thisturnedout tosolveaconjectureinastronomyregardinganinstanceoftheMainProblem.In[71] jointworkwithLudwigKuznia,weinvestigatedthecasewhen r z isaBlaschke product.Theresultingsharpboundis n +3andtheproofissimple.Thisapplies togravitationallensesconsistingofcollinearpointmasses. In[38],C.D.FassnachtandC.R.KeetonastrophysicistsandD.Khavinson posedanotherinstanceoftheMainProblemandreducedittothefollowingsimple problemincomplexanalysis: Problemi: Giveanupperboundforthenumberofsolutionstothefollowing transcendentalequation,where k isarealparameterand w isacomplexparameter, withtheprincipalbranchof arcsin arcsin k z + w = z; .3.5 11

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Solutionstothisequationrepresentpositionsofimageslensedbyanelliptical galaxywithanisothermal"density.Astronomershaveobserveduptofoursuch imagesofasinglesource,butitisnotevenobvious,ataglance,thatthenumberof solutionsto.3.5isnite.Thefunctionappearingisaharmonicmap,forwhichthe incrementoftheargumentcountsorientation-reversingzeroswithoppositesign,by itselfonlygivinga lower boundonthetotalnumberofzeros.Invertingtheequation, itcanthenbeformulatedasaxedpointproblemconnedtoastrip.Self-composing givesacomplex-analyticxedpointproblem,butthefunctionhasinnitelymany essentialsingularities.In[58]jointworkwithD.Khavinson,weappliedarecent resultfromcomplexdynamics[11]toboundthenumberofattractingxedpoints. Thisgivesanupperboundfororientation-preservingzerosoftheoriginalharmonic map,theimportantstepinobtainingthetotalestimateinourtheorem. Theorem1.3.1 Thenumberofsolutionstotheequation arcsin k z + w = z isboundedby8. W.BergweilerandA.Eremenko[18]improvedthisresulttoaboundof6and foundanexampleahighlyeccentricellipticalgalaxythatattains6.Thiswasa surprisetotheastronomerswhoonlyfoundupto4usingamodelwithunbounded density,ofwhichEq..3.5representsatruncation.Thisurgesthequestion,Are thesiximagesinthetruncatedmodelthevalidconsequenceofacompactlysupported densityoraretheyanartifactofthesharpedge?"Tomakethequestionmoreprecise, takethesimplestapproachforremovingthesharpedgejumpdiscontinuity.Namely, subtractaconstanttomakethedensitycontinuous.Thisintroducesanalgebraicterm intoEq..3.5.Thenwehavethefollowingproblem: Problemii: Aretherechoicesofparametersforwhichthefollowingequationhas 12

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6 solutions? c k arcsin k z + w )]TJ/F19 11.9552 Tf 12.664 0 Td [( z )]TJ/F19 11.9552 Tf 14.349 0 Td [( w + p k 2 )]TJ/F19 11.9552 Tf 11.956 0 Td [( z )]TJ/F19 11.9552 Tf 14.349 0 Td [( w 2 = z; .3.6 Anarmativeanswerwouldbecertaintoinspirefurtherdiscussionamong mathematiciansandastronomersandwouldperhapsleadtoreevaluationofthemainstreammodels.Ifthereisanexamplewith6imagesthenndingitshouldbefeasible, butprovingaboundforthenumberofsolutionstoEq..3.6appearstobeamuch moredicultcasethanEq..3.5.AnothernaturalchangetomaketoEq..3.5 istoincludeatidalforcealinearperturbationresultinginthefollowingversionof theproblem. Problemiii: Giveanupperboundforthenumberofsolutionstothefollowing equation,where k isarealparameterand and w arecomplexparameters,withthe principalbranchof arcsin: arcsin k z + w + z = z: .3.7 Anempiricalinvestigationsuggestsasharpupperboundof8,butincluding thissimplelinearperturbationresistsarigorousproofofevenacrudeupperbound. Thedierentapproachesusedin[18]and[58]eachseemtobreakdown,unlessthe tidalforceisalignedwithoneoftheaxesoftheellipticalgalaxy. 13

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2SingularitiesoftheSchwarzpotentialandLaplacianGrowth Thischapterconsistsofthepaper[78],whichhasbeenacceptedforpublicationin JournalofPhysicsA:MathematicalandTheoretical. TheSchwarzfunctionhasplayedanelegantroleinunderstandingandingeneratingnewexamplesofexactsolutionstotheLaplaciangrowthorHele-Shaw problemintheplane.Theguidingprincipleinthisconnectionisthefactthatnonphysical"singularitiesintheoildomain"oftheSchwarzfunctionarestationary,and thephysical"singularitiesobeysimpledynamics.Wegiveanelementaryproofthat thesameholdsinanynumberofdimensionsfortheSchwarzpotential,introducedby D.KhavinsonandH.S.Shapiro[62]89.Ageneralizationisalsogivenforthe so-calledellipticgrowth"problembydeningageneralizedSchwarzpotential. Newexactsolutionsareconstructed,andwesolveinverseproblemsofdescribingthedrivingsingularitiesofagivenow.Wedemonstrate,byexample,how C n -techniquescanbeusedtolocatethesingularitysetoftheSchwarzpotential. Oneofourmethodsistoprolongavailablelocalextensiontheoremsbyconstructing globalizingfamilies". 2.1LaplacianGrowth Aone-parameterfamilyofdecreasingdomains, f t g ,in R n solvestheLaplacian growthproblemwithsinkat x 0 2 t ifthenormalvelocity, v n oftheboundary )]TJ/F28 7.9701 Tf 7.314 -1.793 Td [(t := @ t isdeterminedbyaharmonicGreen'sfunction, P x ;t ,of t asfollows. 14

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8 > > > > > < > > > > > : v n j )]TJ/F29 5.9776 Tf 5.289 -0.996 Td [(t = r P P =0,in t P j )]TJ/F29 5.9776 Tf 5.288 -0.996 Td [(t =0 P x x 0 ;t )]TJ/F27 11.9552 Tf 21.918 0 Td [(Q K x )]TJ/F24 11.9552 Tf 11.956 0 Td [(x 0 ; .1.1 where K isthefundamentalsolutionoftheLaplaceequation,and Q> 0typically constantdeterminesthesuctionrateat x 0 .Wecanalsoconsider Q< 0forthecase ofasource x 0 whereinjectionoccurs,butthisproblemisstableapproachingasphere inthelimitandissometimescalledthebackward-timeLaplaciangrowth". Thisisa nonlinear movingboundaryproblem,ubiquitousasanidealmodel oratleast,rstapproximationofmanygrowthprocessesinnatureandindustry.We stressthatweareconsideringheretheill-posedzerosurface-tensioncase,wherethe interfacecanencounteracusp.Thezerosurface-tensioncasehasattractedwideand growingattentionmainlyfortworeasonstobebrief:iithasdirectconnections tomanyotherareassuchasclassicalpotentialtheory,integrablesystems,soliton theory,andrandommatrices;iiitadmitsamiraculouscompletesetofexplicit exactsolutionsinthetwo-dimensionalcase. Ifthedomains t arebounded,with Q> 0problem.1.1actuallyproduces a shrinking boundary.Wegetagrowthprocessif t containsinnity,so P thensolves an exterior Dirichletproblem.Insuchasituation,itiscommontoplacethesinkat innitybyprescribingasymptoticsfor r P sothattheuxacrossneighborhoodsof innityisproportionalto Q .Inthetwo-dimensionalcasethiscanberealizedinthe laboratoryusinga Hele-Shawcell .Twosheetsofglassareplacedclosetogetherwitha viscousuidoil"llingthevoidbetweenthem.Asmallholeisdrilledinthecenter ofthetopsheetandaninvisciduidwater"ispumpedinataconstantrate.Then problem.1.1servesasanidealmodelfortheboundaryofthegrowingbubbleof water.Theharmonicfunction P x ;t ,inthiscase,correspondstothe pressure inthe oildomain.Inotherphysicalsettingsmodeledby.1.1, P x ;t canbeaprobability, aconcentration,anelectrostaticeld,oratemperature.Becauseofthehugeamount 15

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ofliterature,wearelimitedtocitinganincompletelistofpapers.Foralistofover 500references,see[45]. Weareparticularlyattractedtothisproblembythelackofexplicitexamples indimensionshigherthantwo.Theexistence,uniqueness,andregularitytheoryare well-developedinarbitrarydimensions,andintheplanethereisanabundanceof explicit,exactsolutions.Indimensionshigherthantwo,theonlyexamplesarea shrinkingsphereinthecasewhentheoildomain" t isboundedortheexterior ofahomotheticallygrowingellipsoidinthecase t isunbounded.Theobvious explanationforthisdeciencyofexplicitexamplesisalackofconformalmapsin higherdimensionsLiouville'sTheoremsinceexactsolutionsareusuallydescribedin termsofatime-dependentconformalmapofthedomaintothedisk.However,exact solutionscanbeunderstoodusingadierenttoolfromcomplexanalysis,theSchwarz functionsee[24]andSection2below.Thefollowingtheoremrelatestothework ofS.Richardson[89]andwasrststatedintermsoftheSchwarzfunctionbyR.F. Millar[80].Also,thediscussiongivenbyS.Howison[48]seemstohaveplayedan importantroleinpopularizingtheuseoftheSchwarzfunctioninstudiesofLaplacian growth. Theorem2.1.1DynamicsofSingularities: R 2 Supposeaone-parameterfamilyofdomains t hassmoothly-changinganalyticboundarywithSchwarzfunction S z;t .ThenitisaLaplaciangrowthifandonlyif @ @t S z;t = )]TJ/F19 11.9552 Tf 9.299 0 Td [(4 @ @z P z;t .1.2 TheSchwarzfunctionisonlyguaranteedtoexistinavicinityofagivenanalytic curve,and apriori thedomainofanalyticityforitstime-derivativeisnotanylarger. Thus,itissurprisingthatforaLaplaciangrowth, @ @t S z;t coincideswithafunctionanalyticthroughout t exceptatthesingularityprescribedatthesink".In otherwords,wecanextractfromequation.1.2thefollowingelegantdescription ofsolutionstoproblem.1.1: Singularitiesin t oftheSchwarzfunctionof @ t donotmoveexceptforonesimplepolestationedatthesink x 0 whichdecreasesin 16

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strengthattherate )]TJ/F27 11.9552 Tf 9.298 0 Td [(Q Sinceequation2.1.2isgiveninphysicalcoordinatesrather thanintroducingauniformizedmathematicalplane",S.Howison[48]hascalledit an intrinsic description.Inrecentpapers,itistypicaltoseeacombinationofthe Schwarzfunctionandtheconformalmapusedtoderivesolutionse.g.[1].Wewill reviewsomefamiliarexamplesinSection4andunderstandthemcompletelyinterms ofTheorem2.1.1. TheSchwarzfunctionhasbeenpartiallygeneralizedbyD.KhavinsonandH. S.ShapirotohigherdimensionsbydeningaSchwarzpotential",asolutionofa certainCauchyproblemfortheLaplaceequation[96].InSection2,wewillreview thedenitionoftheSchwarzfunctionandtheSchwarzpotentialbeforeprovingthe n -dimensionalversionofTheorem2.1.1.Wealsogiveafurthergeneralizationtothe ellipticgrowthproblem.TherestofthepaperisguidedbyTheorem2.2.2,which identies,asthemainobstacle,theproblemofdescribinggloballythesingularitiesoftheSchwarzpotential.InSection4wefollowtheobservationmadebyL. KarpthattheSchwarzpotentialoffour-dimensional,axially-symmetricsurfacescan becalculatedexactly[53].Wegivesomeexplicitexamplesandalsodescribesome examplesofellipticgrowth.InSection5,weuse C n techniquestounderstandthe Schwarzpotential'ssingularitysetforanontrivialexamplein R n includingtheimportantcase n =3.InSection6,wediscusstheconnectiontoquadraturedomains andRichardson'sTheorem. 2.2DynamicsofSingularities 2.2.1TheSchwarzPotential Suppose)-469(isanon-singular,real-analyticcurveintheplane.ThentheSchwarz function S z isthefunctionthatiscomplex-analyticinaneighborhoodof)-423(and coincideswith z on)-312(see[24]forafullexposition.If)-312(isgivenalgebraicallyasthe zerosetofapolynomial P x;y ,wecanobtain S z bymakingthecomplex-linear changeofvariables z = x + iy z = x )]TJ/F27 11.9552 Tf 12.517 0 Td [(iy ,andthensolvingfor z intheequation 17

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P z + z 2 ; z )]TJ/F25 7.9701 Tf 7.077 0 Td [( z 2 i =0.Forinstance,suppose)-454(isthecurvegivenalgebraicallybythe solutionsetoftheequation x 2 + y 2 2 = a 2 x 2 + y 2 +4 2 x 2 C.Neumann'soval". Thenchangingvariableswehave z z 2 = a 2 z z + 2 z + z 2 .Solvingfor z gives S z = z a 2 +2 2 + z p 4 a 4 +4 a 2 2 +4 2 z 2 2 z 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(" 2 Suppose)-377(ismoregenerallyanonsingular,analytic hypersurface in R n ,and considerthefollowingCauchyproblemposedinthevicinityof.Thesolutionexists andisuniquebytheCauchy-KovalevskayaTheorem. 8 > > > < > > > : w =0near)]TJ/F27 11.9552 Tf -21.766 -20.922 Td [(w j )]TJ/F19 11.9552 Tf 9.107 1.793 Td [(= 1 2 jj x jj 2 r w j )]TJ/F19 11.9552 Tf 9.107 1.794 Td [(= x .2.3 Denition2.2.1 Thesolution w x oftheCauchyproblem2.2.3iscalledtheSchwarz Potentialof. Example: Let)-384(:= f x 2 R n : jj x jj 2 = r 2 g beasphereofradius r .When n =2, itiseasytoverifythat w z = r 2 log j z j +1 = 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(log r solvestheCauchyProblem .2.3,andinhigherdimensionstheSchwarzpotentialis w x = )]TJ/F28 7.9701 Tf 31.724 4.707 Td [(r n n )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 jj x jj n )]TJ/F26 5.9776 Tf 5.756 0 Td [(2 + n 2 n )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 r 2 In R 2 ,theSchwarzfunctioncanbedirectlyrecoveredfromtheSchwarzpotential.Consider S z =2 @ z w = w x )]TJ/F27 11.9552 Tf 11.494 0 Td [(iw y .TheCauchy-Riemannequationsfor S follow fromharmonicityof w ,and r w = x on)-327(implies S z = z on. ThisgivesapartialgeneralizationoftheSchwarzfunction.ThereectionprincipleassociatedwiththeSchwarzfunctiondoesnotgeneralizetohigherdimensionsby thisoranyothermeans,buttheSchwarzpotentialretainsotherdesirableproperties. Inparticular,itallowsustogeneralizeTheorem2.1.1tohigherdimensions. 2.2.2LaplaciangrowthandtheSchwarzpotential ThefollowingtheoremgeneralizesThereom2.1.1.Weconsiderafamilyofdomains t R n sothat t hasananalyticboundarywithanalytictime-dependence.Such 18

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aregularityassumptionisnaturalforussinceweareinpursuitofexplicit,exact solutions.However,weshouldmentionthatanalyticityoftheboundaryisanecessaryconditionforexistenceofaclassicalsolution,andmoreoverforananalytic initialboundary,thereexistsauniquesolutionremaininganalyticwithanalytictimedependenceforatleastsomeintervaloftimesee[37]and[104].Let w x ;t denote theSchwarzpotentialoftheboundary)]TJ/F28 7.9701 Tf 208.267 -1.794 Td [(t of t Theorem2.2.2DynamicsofSingularities: R n If t and w x ;t areasabove then t solvestheLaplaciangrowthproblem.1.1ifandonlyif @ @t w x ;t = )]TJ/F27 11.9552 Tf 9.298 0 Td [(nP x ;t .2.4 where n isthespatialdimension.Inparticular,singularitiesoftheSchwarzpotential intheoildomain"donotdependontime,exceptforonestationedatthesource sinkwhichdoesnotmovebutsimplychangesstrength. Remark1: Thisrelatesthesolutionofamathematically-posed"Cauchyproblemto thatofaphysically-posed"Dirichletproblem. Remark2: ConsideringtherelationshipbetweentheSchwarzpotentialandSchwarz function,inthecaseof n =2,theTheoremsaysthat S t = @ @t @ z w = )]TJ/F19 11.9552 Tf 9.299 0 Td [(4 @ z P which isthecontentofTheorem2.1.1. Remark3: ThisiscloselyrelatedtothecelebratedRichardson'sTheorem[89].Actually,theconnectioncanbeestablishedthroughtherolethattheSchwarzpotential playsinthetheoryofquadraturedomainsseeSection2.5.Hereweareabletogive amoreelementaryproofconsistingoftwoapplicationsofthechainrule. Proof. Assume f t g solvestheLaplaciangrowthproblem.Wewillshowthatforeach t w t x ;t and )]TJ/F27 11.9552 Tf 9.298 0 Td [(nP x ;t solvethesameCauchyproblem.Thenbytheuniqueness partoftheCauchy-KovalevskayaTheorem,theyareidentical. 19

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First,wewillshowthat w t x ;t j )]TJ/F29 5.9776 Tf 5.289 -0.996 Td [(t =0.Considerapoint x t whichison)]TJ/F28 7.9701 Tf 78.556 -1.793 Td [(t attime t .Thechainrulegives d dt w x t ;t = r w x t ;t _x t + w t x t ;t .2.5 Ontheotherhand,bytherstpieceofCauchydatain.2.3, d dt w x t ;t = d dt 1 2 jj x t jj 2 = x t _x t .2.6 BythesecondpieceofCauchydata, x t _x t = r w x t ;t _x t .2.7 Combining.2.6and.2.7withequation.2.5gives w t x ;t j )]TJ/F29 5.9776 Tf 5.288 -0.996 Td [(t =0.2.8 Wearedoneifwecanshowthat r w t j )]TJ/F29 5.9776 Tf 5.289 -0.996 Td [(t = )]TJ/F27 11.9552 Tf 9.299 0 Td [(n r p .Givensomeposition, x ,let T x assignthevalueoftimepreciselywhentheboundary,)]TJ/F28 7.9701 Tf 304.608 -1.793 Td [(t ,ofthegrowingdomainpasses x .ThenbytheCauchydatadeningtheSchwarzpotential.2.3, w x k x 1 ;x 2 ;:::;x n ;T x 1 ;x 2 ;:::;x n = x k .Takingthepartialwithrespectto x k ofthe kth equationgives w tx k T x k + w x k x k =1.Summingthese k equationstogethergives r w t r T + w = n: .2.9 Since)]TJ/F28 7.9701 Tf 37.983 -1.794 Td [(t isthelevelcurve T x = t r T isorthogonalto)]TJ/F28 7.9701 Tf 92.232 -1.794 Td [(t ,and r T = v n jj v n jj 2 ,where v n isthenormalvelocityof)]TJ/F28 7.9701 Tf 135.47 -1.793 Td [(t .Recall, v n = r P .Thus, r T = r P jjr P jj 2 .Substitution intoequation.2.9gives r w t r P = )]TJ/F27 11.9552 Tf 9.299 0 Td [(n jjr P jj 2 Byequation.2.8, w t j )]TJ/F29 5.9776 Tf 5.288 -0.997 Td [(t =0,whichimpliesthat r w t and r P areparallel. So, r w t j )]TJ/F29 5.9776 Tf 5.288 -0.996 Td [(t = )]TJ/F27 11.9552 Tf 9.299 0 Td [(n r P 20

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2.2.3ACauchyproblemconnectedtoEllipticGrowth AnaturalgeneralizationoftheLaplaciangrowthproblemistoallowanon-constant ltrationcoecient" andporosity" .TheninsteadofLaplace'sequationthe pressuresatisesdiv r P =0andshouldhaveasingularityatthesinkofthesame typeasthefundamentalsolutiontothisellipticequation.Moreover,theDarcy'slaw determiningtheboundaryvelocitybecomes v n = )]TJ/F27 11.9552 Tf 9.299 0 Td [( r P .Fordetails,see[59],[75], [76].Physically,thismodelstheprobleminanon-homogeneousmediumandalso relatestothecaseofHele-Shawcellsoncurvedsurfacesintheabsenceofgravity studiedin[105,Ch.7]. Wecanformulateanequationsimilartoequation.2.4thatrelatesthepressurefunctionofanellipticgrowthtothetime-dependenceofthesolutiontoacertain Cauchyproblem.Let q x beasolutionofthePoissonequation, div r q = n; .2.10 where n isthespatialdimension.Recallthatasolution q canbeobtainedbytakingthe convolutionof withthefundamentalsolutionofthehomogeneousellipticequation ifoneexists.Weassociatewithanellipticgrowthhavingltration andporosity ,thesolution u ofthefollowingCauchyproblem. 8 > > > < > > > : div r u =0near)]TJ/F27 11.9552 Tf -51.969 -20.922 Td [(u j )]TJ/F19 11.9552 Tf 9.108 1.793 Td [(= q r u j )]TJ/F19 11.9552 Tf 9.107 1.793 Td [(= r q .2.11 Wecanthinkof u asageneralizedSchwarzpotential".Wehavethefollowingdirect generalizationofTheorem2.2.2.AsinSection2.2,assume t hasanalyticboundary withanalytictime-dependence. Theorem2.2.3 If )]TJ/F28 7.9701 Tf 7.314 -1.793 Td [(t = @ t and u x ;t isthesolutionto2.2.11posedon )]TJ/F28 7.9701 Tf 7.314 -1.793 Td [(t then t 21

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isanellipticgrowthwithpressurefunction P x ;t ifandonlyif @ @t u x ;t = )]TJ/F27 11.9552 Tf 9.299 0 Td [(nP x ;t .2.12 Proof. AsintheproofofTheorem2.2.2,weshowthatbothsidesof2.2.12solvethe sameCauchyproblem. Therstpartoftheargumentissimilarinshowingthat. u t x ;t j )]TJ/F29 5.9776 Tf 5.289 -0.996 Td [(t =0.2.13 Considerapoint x t whichison)]TJ/F28 7.9701 Tf 77.29 -1.793 Td [(t attime t .Thechainrulegives d dt u x t ;t = r u x t ;t _x t + u t x t ;t .2.14 Ontheotherhand,bytherstpieceofCauchydatain.2.11andthechainrule again, d dt u x t ;t = d dt q x t = r q x t _x t .2.15 BythesecondpieceofCauchydata, r q x t _x t = r u x t ;t _x t .2.16 Combining.2.15and2.2.16withequation.2.14givestheequation.2.13. Wearedoneifwecanshowthat r u t j )]TJ/F29 5.9776 Tf 5.288 -0.997 Td [(t = )]TJ/F27 11.9552 Tf 9.299 0 Td [(n r P Weagainlet T x assignthevalueoftimewhen)]TJ/F28 7.9701 Tf 178.008 -1.793 Td [(t passes x .Thenbythe Cauchydatadening u r u x ;T x = r q .Multiplybothsidesby andtakethe divergence: r u t r T +div r u =div r q ; .2.17 which,bydenitionof u and q ,simpliesto r u t r T = n: .2.18 22

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AsintheproofofTheorem2.2.2, r T = v n jj v n jj 2 ,where v n isthenormalvelocityof)]TJ/F28 7.9701 Tf 135.455 -1.793 Td [(t exceptnow v n = )]TJ/F27 11.9552 Tf 9.299 0 Td [( r P .Thus, r T = r P jjr P jj 2 .Substitutionintoequation.2.18 gives r u t r P = )]TJ/F27 11.9552 Tf 9.299 0 Td [(n jjr P jj 2 Byequation.2.13, u t j )]TJ/F29 5.9776 Tf 5.288 -0.997 Td [(t =0,whichimpliesthat r u t and r P areparallel. So, r u t j )]TJ/F29 5.9776 Tf 5.289 -0.996 Td [(t = )]TJ/F27 11.9552 Tf 9.298 0 Td [(n r P Letusdiscussaspecialcaseoftheabove.SupposethattheProblem.2.10 hasasolution q thatisentire.Let denote ,andsuppose isalsoentire.For instance, = 1 x 2 +1 and = x 2 +1gives =1and q = x 4 + x 2 .When =1asinthis case,theellipticgrowth"isjustaLaplaciangrowthwithavariable-coecientlaw governingtheboundaryvelocity.Theproblem.2.11dening u becomesaCauchy problemforLaplace'sequationwithentiredata.ThisistherealmoftheSchwarz potentialconjectureformulatedbyKhavinsonandShapiro: Conjecture2.2.4Khavinson,Shapiro Suppose u solvestheCauchyproblemfor Laplace'sequationposedonanonsingularanalyticsurface )]TJ/F44 11.9552 Tf 12.639 0 Td [(withreal-entiredata. Thenthesingularitysetof u iscontainedinthesingularitysetoftheSchwarzpotential w Theconjectureholdsintheplaneandhasbeenshowntoholdgenerically" inhigherdimensions[94].Iftheconjectureistrue,thenforthecasewhen =1, thesingularitiesof u arecontrolledthroughouttimebythoseof w .Combiningthis withTheorems2.2.2and2.2.3impliesthat,givenasolutionoftheLaplaciangrowth problem.1.1,theexactsameevolutioncanbegeneratedamidanellipticgrowth lawwith =1byapressurefunctionhavingsingularitiesatthesamelocationsas thoseof w .Thesingularitiesmayhavedierenttime-dependenceandbeofadierent type. Forinstance,considertheplaneandthesimplestLaplaciangrowthofsuction fromthecenterofacirclesothatattime t ,theSchwarzfunctionis 1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(t z aconstant rateofsuction.Letusdeterminethepressurerequiredtogeneratethesameprocess 23

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when = 1 x 2 +1 and = x 2 +1.Tosolvefor u ,wenoticethat @ z u isanalyticand coincideswith2 x 3 + x ontheshrinkingcircle.Since x = z + S z 2 ontheboundary, wehave @ z u = z + 1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(t z 3 = 4+ z= 2+ 1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(t 2 z .Thisiseventrueotheboundarysince bothsidesareanalytic.Thesingulartermsare 5 )]TJ/F25 7.9701 Tf 6.586 0 Td [(8 t +3 t 2 4 z and 1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(3 t +3 t 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(t 3 4 z 3 .Thus,in ordertogeneratethesamemovie",thepressuremusthaveafundamentalsolution typesingularityalongwithaweakmulti-pole"attheorigin,bothdiminishingat non-constantrates. 2.3Examples InthissectionwewillexplainsomeexplicitsolutionsintermsoftheTheorems2.1.1, 2.2.2,and2.2.3. 2.3.1Laplaciangrowthintwodimensions Firstwereviewsomefamiliarexamplesintheplane,wheretypicallyatime-dependent conformalmapisintroduced.Instead,weworkentirelywiththeSchwarzfunction andcheckthatTheorem2.1.1issatised. Example1: Considerthefamilyofdomains D withboundarygivenbythe curves f z : z = aw 2 + bw; j w j < 1 g with a;b real.TheSchwarzfunctionisgiven by S z = )]TJ/F19 11.9552 Tf 9.299 0 Td [(2 ab= a )]TJ 12.414 9.876 Td [(p a 2 +4 bz +4 b 3 = a )]TJ 12.414 9.876 Td [(p a 2 +4 bz 2 whichhasasingle-valued branchintheinteriorofthecurveforappropriateparametervalues a and b .Theonly singularitiesoftheSchwarzfunctioninteriortothecurveareasimplepoleandapole ofordertwoattheorigin.Givenaninitialdomainfromthisfamilywecanchoose aone-parametersliceofdomainssothatthesimplepoleincreasesresp.decreases whilethepoleofordertwodoesnotchange.Thisgivesanexactsolutiontothe Laplaciangrowthproblemwithinjectionresp.suctiontakingplaceattheorigin. Inthecaseofinjection,thedomainapproachesacircle.Inthecaseofsuction,the domaindevelopsacuspinnitetime. Insteadofjustonesinkorsource x 0 withrate Q ,letusextendproblem2.1.1 byallowingformultiplesinksand/orsources x i withsuction/injectionrates Q i .This 24

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Figure2.1:Anexamplewithtwosinks. istheformulationoftheproblemwhichisoftenmade,forinstance,seetheexcellent exposition[105].TheproofofTheorem2.2.2carriesthroughwithoutchangessothat thetime-derivativeoftheSchwarzpotentialstillcoincideswith )]TJ/F27 11.9552 Tf 9.299 0 Td [(nP x ;t .Theonly dierenceisthatnowtherecanbemultipletime-dependentpoint-singularitiesinside t Example2: WerstconsiderthefamilyofcurvesmentionedinSection2.1. TheSchwarzfunctionoftheboundaryis S z = z a 2 +2 2 + z p 4 a 4 +4 a 2 2 +4 2 z 2 2 z 2 )]TJ/F27 11.9552 Tf 11.956 0 Td [(" 2 whichhastwosimplepolesat z = eachwithresidue a 2 +2 2 = 2.Inorderto satisfytheconditionsimposedbyTheorem2.2.2wechoose =1tobeconstant. Thenwechoose a t tobedecreasingincreasingtoobtainsuctioninjectionattwo sinkssources.Inthecaseofsuction,theovalformsanindentationatthetopand bottomandbecomesincreasinglypinchedastheboundaryapproachestwotangent circlescenteredat 1,thepositionsofthesinksseeFigure2.1. Forthenextexample,weconsiderthecaseofProblem.1.1wheretheoil domain" t isunboundedwithasinkatinnity. 25

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Example3: WerecalltheSchwarzfunctionforanellipsegivenbythesolution setoftheequation x 2 a 2 + y 2 b 2 =1.Changingvariableswehave z + z 2 a 2 )]TJ/F25 7.9701 Tf 11.09 5.698 Td [( z )]TJ/F25 7.9701 Tf 7.077 0 Td [( z 2 b 2 =4.Solving for z gives S z = a 2 + b 2 a 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(b 2 z + 2 ab b 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(a 2 p z 2 + b 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(a 2 S z hasasquarerootbranchcut alongthesegmentjoiningthefoci p a 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(b 2 ,butweareonlyinterestedintheexterior oftheellipse,where S z isfreeofsingularities.Thisalreadyguaranteesthatany evolutionofellipsesthathasanalytictimedependencecanbegeneratedbypreparing thecorrectasymptoticpressureconditionstomatch S t z;t whichisonlysingularat innity.Inotherwords,wecanuseequation2.1.2toworkbackwardsinspecifyingthe pressureconditiontogeneratethegivenow.Sincetherearenonitesingularities, weonlyhavetospecifytheconditionsatinnity.Arealisticcaseisiftheasymptotic conditionissteadyandisotropic: S t z !1 ;t k=z foraconstant k independent of t .Takeahomotheticgrowthwith a t = a p t and b t = b p t fromsomeinitial ellipsewithsemi-axes a and b .Then S z;t = a 2 + b 2 a 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(b 2 z + 2 ab b 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(a 2 p z 2 + t b 2 )]TJ/F27 11.9552 Tf 11.956 0 Td [(a 2 ,andwe have S t z;t = k 1 p z 2 + t b 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(a 2 ,where k =2 ab 2.3.2Examplesandnon-examplesin R 4 Next,weconsideraxially-symmetric,four-dimensionaldomains.Thisturnsouttobe simplerthanthemorephysicallyrelevant R 3 ,andwewillseeinthenextsubsection thatitisisequivalenttocertaincasesofellipticgrowthintwoandthreedimensions. LaviKarp[53]hasgivenaprocedure,includingseveralexplicitexamples,forobtaining thesingularitiesoftheSchwarzpotentialforadomainthatistherotationinto R 4 ofadomainin R 2 withSchwarzfunction S z .Weoutlineherethisprocedurefor ndingtheSchwarzpotential w x 1 ;x 2 ;x 3 ;x 4 .Since w solvesaCauchyproblemfor axiallysymmetricdataposedonanaxiallysymmetrichypersurface,with,say x 1 as theaxisofsymmetry,itcanberegardedasafunctionoftwovariables.Write x = x 1 ;y = p x 2 2 + x 2 3 + x 2 4 ,and w x 1 ;x 2 ;x 3 ;x 4 = U x;y .Whatmakes R 4 convenientto workwithisthefactthat V x;y = y U x;y isaharmonicfunctionofthevariables x and y .Thus,nding U x;y isreducedtosolvinganalgebraicCauchyproblemin theplane,whichcanbedoneintermsoftheSchwarzfunction S x + iy = S z .The 26

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stepsforwritingthissolutionareoutlinedbelow. Step1: Write f z = i 2 S z S z )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 z Step2: Findaprimitivefunction F z for f z Step3: Write V x;y = Re f F z g .ThentheSchwarzpotentialforis U x;y = V x;y + const: y Oneoftheexamplescarriedthroughthisprocedurein[53]isthefamilyof limacons"fromExample1.TheresultisthattheSchwarzpotentialcanbeexpandedabouttheoriginas w x 1 ;x 2 ;x 3 ;x 4 = A 2 a;b @ @x 1 2 j x j )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 + A 1 a;b @ @x 1 j x j )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 + A 0 a;b j x j )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 + H x ,where H x isharmonicand A 2 a;b = )]TJ/F27 11.9552 Tf 9.299 0 Td [(b 2 a 4 = 12, A 1 a;b = ba 2 a 2 +2 b 2 = 2,and A 0 a;b = )]TJ/F19 11.9552 Tf 9.299 0 Td [( a 4 +6 a 2 b 2 +2 b 4 = 2.Wecaninterpretone-parameter slicesofthisfamilyasaLaplaciangrowthifwefurtherextendproblem2.1.1toallow formulti-poles"see[36]foradiscussionofmulti-polesolutionsintheplane.If wewantaLaplaciangrowthwithjustasimplesinkthenaccordingtothedynamicsof-singularitiesimposedbyTheorem2.2.2,weneedtochoosethetime-dependence of a and b sothattheonlysingularitywhosecoecientchangesisthefundamentalsolutiontypesingularity A 0 a;b j x j )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 .Thus,where C 1 C 2 areconstants,weneedto have: 8 < : A 2 a t ;b t = C 2 A 1 a t ;b t = C 1 .3.19 Unfortunately,solutions a and b ofthissystemarelocallyconstantsothat A 0 must thenbeconstantandthewholesurfacedoesnotmoveatall.Theotherexamples ofaxiallysymmetricdomainsconsideredin[53]alsorequireintroducingmulti-poles orevenacontinuumofsingularities,otherwisetheconditionsimposedbysimple sources/sinksleadstoasimilarlyoverdeterminedsystem.Roughlyspeaking,thedifcultyisthat f z fromStep1abovegenerallyhasmoresingularitiesthan S z .Thus, ifaclassofdomainsintheplanehasenoughparameterstocontrolthesingularities andobtainaLaplaciangrowth,thenrotationinto R 4 introducesmoresingularities whichmustbecontrolledwiththesamenumberofparameters. Thereareexceptions:wecandescribesomeexactsolutionsinvolvingasimple 27

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sourceandsinknomultipoles.Considerthehypersurfacesofrevolutionobtained byrotationfromafamilyofcurveswhoseSchwarzfunctionshavetwosimplepoles at z = 1withnotnecessarilyequalresidues.Thisisatwo-parameterfamily ofsurfaces;asparameters,wecantaketheresiduesoftheSchwarzfunctionofthe prolecurves.Letdenotethedomainintheplaneboundedbytheprolecurve. TheSchwarzfunctionhastheform S z = A z )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 + B z +1 + c A;B + d A;B z + z 2 H z;A;B ; where H z;A;B isanalyticin.Inwhatfollows,wewillsuppressthedependence on A and B ofhigher-degreecoecients.FollowingSteps1through3above,wehave f z = i 2 S z S z )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 z = i 2 A 2 z )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 2 + B 2 z +1 2 + C 1 z )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 )]TJ/F27 11.9552 Tf 19.718 8.087 Td [(C 2 z +1 + H 1 z ; where H 1 z isanalyticin.ThenforStep2weneedaprimitivefunctionfor f z whichis F z = i 2 A 2 z )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 + B 2 z +1 + C 1 Log z )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(C 2 Log z +1+ H 2 z ; where H 2 z isanalyticin. ThenforStep3wehave V x;y = Re f F z g = A 2 y x )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 2 + y 2 + B 2 y x +1 2 + y 2 + C 1 arg z )]TJ/F19 11.9552 Tf 11.955 0 Td [(1+ C 2 arg z +1+ H 3 z Ifwecanvary A and B inawaythatkeeps C 1 and C 2 constant,thenthetimederivativeoftheSchwarzfunctionwillsatisfythedynamics-of-singularitiescondition. Thisseemsatrsttobeanotheroverdeterminedproblem,butactually C 1 and C 2 must beequal!Otherwise,thetwobranchcutsof C 1 arg z )]TJ/F19 11.9552 Tf 11.988 0 Td [(1and C 2 arg z +1willnot canceleachotheroutsidetheinterval[ )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1],andtheSchwarzpotentialwillbecome singularonthesurfaceitself.Thiscannothappensincethesurfacehasnopoints in R n thatarecharacteristicfortheCauchyproblem.Therefore,since C 1 and C 2 28

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Figure2.2:Aproleofanaxially-symmetricsolutionin R 4 withinjectionatonepointand suctionatanother.Theinitialcurveisplottedinbold. areequal,wespendonlyonedimensionofourparameterspacecontrollingthenonphysical"segmentofsingularities.Thisleavesfreedomforthephysical"singularities tomove,atleastlocally,alongaone-dimensionalsubmanifoldofparameters.Figure 2.2showstheevolutionoftheprolecurveforatypicalexamplethatcanbeobtained inthisway. Weomitthecumbersomeformulaeforthetime-dependenceofcoecientsin thealgebraicdescriptionofsuchexactsolutions.Thetwo-parameterfamilyofhypersurfacesfromwhichtheyareselectedcanbedescribedbythesolutionsetof: x 1 + h 2 a 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(h 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(h x 1 + h 2 a 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(h 2 2 + x 2 2 + x 2 3 + x 2 4 2 a 2 )]TJ/F19 11.9552 Tf 10.494 8.088 Td [( a 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(h 2 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(a 2 x 2 2 + x 2 3 + x 2 4 a 2 a 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(h 2 = x 1 + h 2 a 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(h 2 2 + x 2 2 + x 2 3 + x 2 4 2 : Similarly,onecanobtainexampleswheretheSchwarzfunctionhasthreeor 29

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moresimplepoles.Againthesuction/injectionrateswillhavetooccurinaprescribed wayorelsethetime-derivativeoftheSchwarzpotentialwillhavesingularsegments whicharediculttointerpretphysically. Remark: Therigidityoftheinter-dependenceofinjection/suctionratesin theaboveexampleismadelessseverebythefactthattheinitialandnaldomains onlydependonthetotalquantitiesinjectedandremovedatthesourceandsink respectively,andtheyareindependentoftheratesandorderofworkofthesource andsinksee[105]:theproofextendswordforwordtohigherdimensions.Thus, injectionandsuctioncanhappeninanymanner,sayoneatatime,andwewilllose themovie"butretainthenaldomain. Inthenextsectionwewillbeinterestedinexamplesthatcorrespondtoaxially symmetricsurfacesthatdonotintersecttheaxisofsymmetry.Forinstance,to generateatorus,wecanchoosetheprolecurvetobeacircleofradius R andcenter ai a>R> 0.TheSchwarzfunctionis S z = R 2 z )]TJ/F28 7.9701 Tf 6.586 0 Td [(ai )]TJ/F27 11.9552 Tf 11.537 0 Td [(ai .Step1gives f z = i= 2 R 2 z )]TJ/F28 7.9701 Tf 6.587 0 Td [(ai )]TJ/F27 11.9552 Tf -424.076 -20.921 Td [(ai R 2 z )]TJ/F28 7.9701 Tf 6.587 0 Td [(ai )]TJ/F27 11.9552 Tf 10.863 0 Td [(ai )]TJ/F19 11.9552 Tf 10.863 0 Td [(2 z .Step2gives F z = )]TJ/F28 7.9701 Tf 6.586 0 Td [(iR 4 2 z )]TJ/F28 7.9701 Tf 6.586 0 Td [(ai +2 R 2 aLog z )]TJ/F27 11.9552 Tf 10.863 0 Td [(ai + H z ,where H z isanalytic.Step3gives V z = )]TJ/F28 7.9701 Tf 6.587 0 Td [(R 4 y )]TJ/F28 7.9701 Tf 6.587 0 Td [(a 2 j z )]TJ/F28 7.9701 Tf 6.586 0 Td [(ai j 2 +2 R 2 a log j z )]TJ/F27 11.9552 Tf 11.061 0 Td [(ai j + R z ,where R z isfree ofsingularities.Finally,thesingularpartof U x;y is R 4 2 ay @ @y 1 x 2 + y 2 + 2 R 2 y log j z )]TJ/F27 11.9552 Tf 11.955 0 Td [(ai j ThiscalculationfortheSchwarzpotentialofthefour-dimensionaltoruswas carriedoutin[2]anddiscussedinconnectionwithaclassicalmean-value-propertyfor polyharmonicfunctions. 2.3.3Examplesofellipticgrowth Examplesofaxially-symmetric,four-dimensionalLaplaciangrowthalsosolvecertain ellipticgrowthproblemsintwoandthreedimensions.Thetwo-dimensionalprole solvestheplanarellipticgrowthproblemwheretheltrationcoecient =1is constant,andtheporosity x;y = y 2 .Indeed,wecancheckthatTheorem2.2.3 issatised.TheSchwarzpotential U x;y ,reducedtotwovariables,satisesthe equation U + 2 U y y =0.Sincediv y 2 r U = y 2 U +2 yU y ,then U solvestheCauchy problem 30

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8 > > > < > > > : div y 2 r U =0near)]TJ/F27 11.9552 Tf -52.509 -20.921 Td [(U j )]TJ/F19 11.9552 Tf 9.108 1.794 Td [(= q r U j )]TJ/F19 11.9552 Tf 9.107 1.793 Td [(= r q .3.20 with q x;y = x 2 + y 2 = 8solvingthePoissonequationdiv y 2 r q = y 2 Thethreedimensionalsurfacesofrevolutiongeneratedbythesameprole curvessolveathree-dimensionalellipticgrowthifwechoose =1againconstantand porosity x;y;z = p y 2 + z 2 Itismostinterestingwhenthedomain,atleastinitially,avoidstheline f y =0 g where x;y vanishes.Consider,forinstance,acircleofradius R centeredat ai .This correspondstothecalculationattheendofSection2.3forthefour-dimensionaltorus. Accordingly,ashrinkingcirclecanbegeneratedbyasimplesourcecombinedwitha dipoleow"positionedatthecenterofthecircle. Asimilarcalculationappliesmoregenerallywhen x;y = y 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(m x;y = y m with m apositiveinteger,andwecanconsidermoregeneraldomainsthancircles.For instance,awell-knownclassicalsolutionoftheLaplaciangrowthintheplaneinvolves domains t conformallymappedfromtheunitdiscbypolynomials.Physicallythe solutionhasasinglesinkpositionedattheimageoftheoriginundertheconformal map.TheSchwarzfunctionofsuchan t ismeromorphicexceptatthesinkwhereits highestorderpolecoincideswiththedegreeofthepolynomial.So, S z = P k i =1 a i z i + H z ,where H z isanalyticin t .Thesolution q ofdiv y 2 r q = y m is q x;y = y m +2 m +2 m +3 .Tosolvefor U x;y werstnoticethat V x;y = yU x;y isharmonicand solvesaCauchyproblemwithdata yq x;y = y m +3 m +2 m +3 .Thus, @ z V = )]TJ/F28 7.9701 Tf 11.17 4.707 Td [(i 2 y m +2 m +2 = )]TJ/F28 7.9701 Tf 11.17 4.708 Td [(i 2 z )]TJ/F28 7.9701 Tf 6.586 0 Td [(S z m +2 m +2 i m +2 canbeanalyticallycontinuedawayfromtheboundary.Asaresult, theowcanbegeneratedbyacombinationofmultipoles"positionedatthesame pointofordernotexceeding k m +2.ThisresemblestheresultofI.Loutsenko[75] statingthatthesameevolutioncanbegeneratedbymultipolesofacertainorder underanellipticgrowthwhere =1constantand = 1 y 2 p ,with p apositiveinteger. Thisfails,inaninterestingway,fornegativevaluesof m .Forinstance,if 31

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Figure2.3:Anellipticgrowthwithmulti-polesoforderupto3positionedat z = i .The Schwarzfunctionhasamovingsingularity. x;y = y 4 and x;y =1 =y 2 ,thenacircleofshrinkingradius R centeredat ai isnotgeneratedbymultipolespositionedat ai .Instead,thegeneralizedSchwarz potential U x;y hassingularitiesatthemovingpoint i p a 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(r 2 .Ifweinsteadallow thecenteroftheshrinkingcircletomoveinawaythatkeeps p a 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(r 2 constant, thentheevolutioncanbegeneratedbymulti-polesatthispointoforderupto3see gure2.3.Toreiterate,forthisevolutionofshrinkingcircleswithmovingcenter, thegeneralizedellipticSchwarzpotentialissingularatastationarypointwhile theanalyticSchwarzfunctionhasamovingsingularity.Suchanexamplehasbeen anticipatedin[59],whereasystemofnonlinearODEswasgivengoverningboththe strengthandthemovingpositionoftheSchwarzfunction'ssingularitiesunderan ellipticgrowth. 2.4TheSchwarzpotentialin C n Theprevioussectionscallforadeeperlookintothesingularitiesthatcanarisefrom Cauchy'sproblemfortheLaplaceequation.Certaintechniquescanonlybeappliedif theproblemiscomplexied".Accordingtothealgebraicformoftheinitialsurface 32

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anddata,wecanalloweachvariabletoassumecomplexvalues.Wethenconsider theCauchyproblemin C n wheretheoriginal,physicalproblembecomesarelatively smallslice.Wecanlooselydescribetheadvantageofa C n -viewpointasfollows: Considerrstthewaveequationin R n .Iftheinitialsurfaceisnon-singularand algebraicandthedataisreal-entire,thenwherecanthesolutionhavesingularities?A singularitycanpropagatetosomepointifthebackwardslight-conefromthispointis tangenttotheinitialsurface.Thesameistrue,atleastheuristically,fortheLaplace equation,exceptthelightcone"emanatingfromapoint x 0 isthe isotropiccone := f P n i =1 z i )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 0 i 2 =0 g ,residingin C n andonlytouching R n at x 0 .Thus,the initialsourceofthesingularityislocatedonthepartofthecomplexiedsurfaceonly visibleiftheproblemisliftedto C n Leray'sprinciple"givesthegeneral,precisestatementoftheabovedescription ofpropagationofsingularities.Itisonlyknowntoberigorouslytrueinaneighborhoodoftheinitialsurface.Intwodimensions,wheretheSchwarzpotentialcanbe calculatedeasily,onecancheckexamplestoseeifLeray'sprinciplegivescorrectglobal resultsitseemsto.Atthesametime,thisgivesanappealinggeometricexplanation"ofthesourceofsingularitiesandrevealsthattheyarethefoci"ofthecurvein thesenseofPluckersee[52,Section1]andthereferencestherein. Inarbitrarydimensions,G.Johnssonhasgivenaglobalproof[52]ofLeray's principleforquadraticsurfaces.AsJohnssonpointsout,amajorstepintheproof reliesonthefactthatthegradientofaquadraticpolynomialislinear,sothata certainsystemofequationscanbeinvertedeasily.Thisbecomesmuchmoredicult forsurfacesofhigherdegree,indeed,perhapsprohibitivelydicultevenforspecic examples. Inthissectionweconsiderafamilyofsurfacesofdegreefour,thesurfacesof revolutiongeneratedbytheNeumannovalsfromExample2inSection2.3.Leray's principlegivesanappealinggeometricexplanation"forthesingularitiesoftheSchwarz potentialinthisexample,butfortherigorousproof,weapplyanadhoccombination ofother C n techniquesactually C 2 ,aftertakingintoaccountaxialsymmetry. 33

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WerequirethefollowingtwolocalextensionTheorems. Theorem2.4.1Zerner Let v beaholomorphicsolutionoftheequation Lv =0 inadomain C n with C 1 boundary,andassumethatthecoecientsof L are holomorphicin .Let z 0 2 @ .If @ isnon-characteristicat z 0 withrespectto L then v extendsholomorphicallyintoaneighborhoodof z 0 Inordertodene non-characteristic forarealhypersurfacegivenbythezero setof ,supposethepolynomial P x ; r expressestheleadingordertermof L .Then )-465(ischaracteristicat p if P x ; r vanishesat x = p .Forinstance,if L isthe Laplacianthentheconditionfor f =0 g tobecharacteristicis P n i =1 2 x i =0. Inordertostatethenexttheoremsee[31]fortheproof, M isahypersurfaceof real codimensiononedividingadomaininto + and )]TJ/F19 11.9552 Tf 7.084 1.793 Td [(.Also, wesupposetheleadingorderpart P Z;D ofthedierentialoperator L factors as P Z;D = A Z Q Z;D ,where A Z isholomorphicin.Let X denotethe everywhere-characteristiczerosetof A Z having complex codimensionone. Theorem2.4.2Ebenfelt,Khavinson,Shapiro Assume M non-characteristic for Q Z;D at p 0 2 M ,andthattheholomorphichypersurface X isnon-singularat p 0 andmeets M transversallyatthatpoint.Thenanyholomorphicsolution v in )]TJ/F44 11.9552 Tf -426.29 -19.128 Td [(of Lv =0 extendsholomorphicallyacross p 0 Theorem2.4.3 Let W x betheSchwarzpotentialoftheboundary )]TJ/F44 11.9552 Tf 11.246 0 Td [(ofthedomain := f x 2 R n : P n i =1 x 2 i 2 )]TJ/F27 11.9552 Tf 12.644 0 Td [(a 2 P n i =1 x 2 i )]TJ/F19 11.9552 Tf 12.644 0 Td [(4 x 2 1 < 0 g .Then W canbeanalytically continuedthroughout n B where B isthesegment f x 1 2 [ )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 1] ;x j =0 for j = 2 ;::;n g Remarki: Intheplane,itiseasilyseenthat W isonlysingularattheendpoints ofthesegmentseeExample2inSection2.3.In R 4 ,itisanexampledonebyL. Karp[53],whoshowedthattheSchwarzpotentialhastwofundamentalsolutiontype singularitiesattheendpointsalongwithauniformjumpinthegradientacrossthe segment. 34

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Remarkii: Thethree-dimensionalconsequenceofthistheoremisthatifwetakethe surfacesofrevolutiongeneratedbytheNeumannovalsinExample2fromSection2.3 thentheresultingevolutionisaLaplaciangrowth"generatedbyapressurefunction havingsomedistributionofsingularitiesconnedtothesegment f x 2 [ )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1] ;y = 0 ;z =0 g .Thisdrivingmechanismisstillratherobscurethough,sointhenext sectionwedescribeanapproximationbynitelymanysimplesinks. Proof. Werstrecallthat W x isreal-analyticinaneighborhoodofeachnonsingular pointoftheinitialsurfacein R n .Indeed,ifthesurfaceisnonsingular, r j )]TJ/F20 11.9552 Tf 10.084 1.793 Td [(6 = 0 sothat jjr jj 2 j )]TJ/F20 11.9552 Tf 9.108 1.794 Td [(6 =0sothat)-316(iseverywherenon-characteristicin R n forLaplace's equationandtheCauchy-KovalevskayaTheoremapplies. Nextwewrite W x 1 ;x 2 ;:::;x n = u x;y where y = p x 2 2 + x 2 3 + ::: + x 2 n and x = x 1 ,andwerecalltheaxially-symmetricreductionofLaplace'sequation: u + n )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 u y y =0.Since u solvesaCauchyproblemforwhichthedataandboundaryare analytic,theproblemcanbeliftedto C 2 .So u x;y canbeviewedastherestriction to R 2 ofthesolution u X;Y ,validfor X and Y eachtakingcomplexvalues. Wemakethelinearchangeofvariables X = z + w 2 Y = z )]TJ/F28 7.9701 Tf 6.586 0 Td [(w 2 i : u zw + n )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 u z )]TJ/F28 7.9701 Tf 6.586 0 Td [(u w z )]TJ/F28 7.9701 Tf 6.587 0 Td [(w = 0.Nextwemakeanotherchangeofvariables z = f w = f ,usingtheconformal map f = R 4 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 R R 2 )]TJ/F27 11.9552 Tf 11.956 0 Td [( 2 fromtheunitdisktotheproleofforappropriatevalueof R whichisNeumann's ovalseeFigure2.4. Write v ; = u f ;f .Then v = u z f ;f f 0 ,andtheequation satisedby v is v f 0 f 0 + n )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 f )]TJ/F28 7.9701 Tf 6.586 0 Td [(f v f 0 )]TJ/F28 7.9701 Tf 18.185 5.865 Td [(v f 0 =0,or f )]TJ/F27 11.9552 Tf 12.882 0 Td [(f v + n )]TJ/F19 11.9552 Tf -424.076 -20.921 Td [(2 f 0 v )]TJ/F27 11.9552 Tf 11.956 0 Td [(f 0 v =0.Uponclearingdenominators,theleading-orderpartofthe operatoris R 4 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 R R 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 R 2 )]TJ/F27 11.9552 Tf 11.956 0 Td [( 2 )]TJ/F27 11.9552 Tf 5.48 -9.684 Td [( R 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [( R 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 @ @ @ @ : .4.21 Afterthesetransformations,wearriveataCauchyproblemposedon f =1 g 35

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Figure2.4:TheconformalmapfromthedisctotheNeumannoval.Thissimpliesthe C 2 geometrybutmakesthePDEmorecomplicated. withdata v =1 = 2 f f v = f f 0 ,and v = f f 0 .Accordingtotheform oftheleading-orderterm2.4.21,thecharacteristicpointsof f =1 g are 1 ; 1, R; 1 =R 1 =R; R Therestrictionof v tothenon-holomorphicset = correspondstothe originalproblem.Since W x wasobservedtobeanalyticneartheinitialsurface, v ; isanalyticina C 2 neighborhoodofthecircle f =1 ; = g ,evenatthe characteristicpoints 1 ; 1.Weanalyticallycontinue v fromeachpointonthis circlealongaradialpathtowardtheorigin.Let P = e i ;e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i .Weconsidertwo cases.Fortherstcase, 6 =0and 6 = ,and v canbecontinueduptotheorigin.For thesecondcase,when =0or= ,theanalyticcontinuationstopsat =R; 1 =R and )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 =R; )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 =R respectively.Thus, v canbeanalyticallycontinuedtothediskminus thesegmentjoiningthesetwopoints.Thistransformsbyinvertingtheconformal maptothestatementwearetryingtoproveabout W .Foreachcaseweconstruct aglobalizingfamilyinasimilarmannertotheproofoftheBony-SchapiraTheorem [19]. CASE1:Suppose 6 =0and 6 = sothat e i ;e )]TJ/F28 7.9701 Tf 6.587 0 Td [(i isnotonthepreimageoftheaxisofsymmetryof.Let0
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theCauchy-KovalevskayaTheorem, v isanalyticinaneighborhoodofeachpointon .Choose 1 > 0smallenoughsothat v isanalyticina 1 -neighborhoodof .Let 0 denotethistubular C 2 domainofanalyticity. For1 T s ,let N 2 T denotethe 2 -neighborhoodofthesegment f tP ; T t 1 g .Sinceforeach1 t s ,thecharacteristicfortheoperator2.4.21lines through tP alsointersect ,thenforasmallenough 2 ,anycharacteristiclinethat intersects N 2 T alsointersects 0 .Let T betheset co 0 [ N 2 T n co 0 [ 0 ,whereco S denotestheconvexhullofaset S Claim2.4.4 Forpointson @ T n @ 0 ,thetangentplaneisasupportinghyperplane for T Proof. [proofofClaim]Bydenition, T co 0 [ N 2 T ,andthesetwosetsshare aboundarynearpoints p 2 @ T n @ 0 .Thetangentplaneat p 2 @ T n @ 0 isalsoa tangentplanefor @ co 0 [ N 2 T .Byconvexity,itmustbeasupportinghyperplane forco 0 [ N 2 T .Itisthenalsoasupportinghyperplaneforthesubset T Let E := f T : v canbeanalyticallycontinuedto T g .Since1 2 E E is non-empty.Wewillshowthat E isbothopenandclosedrelativeto[ s; 1]andis thereforeequalto[ s; 1].Thefactthat E isclosedfollowsfromthefactthatthe domains T arecontinuousandnested.Toseethat E isopen,weapplyZerner's Theorem.Suppose T 2 E ,i.e., v extendsto T .BytheClaim,thetangentplane P to T at p 2 @ T n @ 0 isasupportinghyperplane.Wemusthavethat P passes through N 2 s .Otherwise, P isasupportinghyperplaneforboth 0 and N 2 s and, therefore,foranysegmentjoiningpointsineachofthesesetsacontradiction.Since P passesthrough N 2 s andnot 0 ,itisnon-characteristic.ByTheorem2.4.1, v extendstoaneighborhoodof p CASE2:Suppose =0or = .Forspecicity,say =0.Then 0 := f r; 1 r ;s r 1 s g passesthroughthecharacteristicpoint ; 1.Wehavealready observed,though,that v isanalyticinaneighborhoodofthepoint ; 1.If s 1 =R 37

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then 0 alsopassesthroughthecharacteristicpoints R; 1 =R ,and =R;R .So,we let s> 1 =R .Thenwecanstillchoosean 1 > 0smallenoughthat v isanalyticin a 1 -neighborhoodof 0 .Weuse 0 againtodenotethisdomainofanalyticity.We canproceedinthesamewayasinthepreviouscase,dening N 2 T and T ,except nowthesetofcharacteristicpoints = intersectstheadvancingboundaryof T for everyvalueof T .Zerner'sTheoremfailsatthispointofintersection,butTheorem 2.4.2appliessincethecomplexline = istransversaltoeachoftheboundaries @ T .Thus,wecanagainprovethattheset E isopenandclosedrelativeto[ s; 1], butrecallthatweassumed s> 1 =R Themethodofproofcanclearlybeappliedtootherexampleshavingaxial symmetry.Inafuturestudy,wehopetoapply C n techniquestosomesurfacesof degreefourthatdonothaveaxial-symmetry,suchasthefamilyofexamplesin R 3 x 2 + y 2 + z 2 2 )]TJ/F19 11.9552 Tf 12.434 0 Td [( a 2 x 2 + b 2 y 2 + c 2 z 2 =0 g ,with a>b>c> 0.ThesearethreedimensionalversionsoftheNeumannovalwithoutaxial-symmetry. 2.5Quadraturedomains Inordertolimitthenumberofdenitionsintheexpositionofourmainresults,we havesofaravoidedexplicitmentionofquadraturedomains",butitwouldberemiss nottodiscussthisimportantconnection.Also,thiswillallowustogiveadetailed approximatedescriptionofthesecondremarkmadeafterthestatementofTheorem 2.4.3. Firstweconsidertheplane.Adomainisa quadraturedomain ifitadmits aformulaexpressingtheareaintegralofanyanalyticfunction f belongingto,say L 1 ,asanitesumofweightedpointevaluationsofthefunctionanditsderivatives. i.e. Z fdA = N X m =1 n k X k =0 a mk f k z m where z i aredistinctpointsinand a mk areconstantsindependentof f 38

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Supposeisabounded,simply-connecteddomainwithnon-singular,analytic boundary.Thenthefollowingareequivalent.Moreover,therearesimpleformulas relatingthedetailsofeach. iisaquadraturedomain. iiTheexteriorlogarithmicpotentialofisequivalenttothatwhichisgenerated bynitelymanyinteriorpointsallowingmultipoles. iiiTheSchwarzfunctionof @ ismeromorphicin. ivTheconformalmapfromthedisktoisrational. Fortheequivalenceofiandiii,see[24,Ch.14].Fortheequivalenceofi, ii,andiv,see[105,Ch.3]. Inhigherdimensions,onesimplyreplacesanalytic"withharmonic"inthe denitionofquadraturedomain.Inconditionii,logarithmic"becomesNewtonian".Inhigherdimensions,multipole"referstoanite-orderpartialderivativeof thefundamentalsolutiontoLaplace'sequation.Inconditioniii,Schwarzfunction" becomesSchwarzpotential",andinsteadofmeromorphic"theSchwarzpotential mustbereal-analyticexceptfornitelymanymultipoles"asdescribedabove.Then theequivalenceofi,ii,andiiipersistsinhigherdimensionssee[62,Ch.4]. Conditionivofcoursedoesnotextend. IftheinitialdomainofaLaplaciangrowthisaquadraturedomain,thenit willstayaquadraturedomainbyvirtueoftheequivalenceofiandiiicombined withTheorem2.2.2.Moreover,accordingtotheformulasomittedhererelatingthe detailsofiandiii,theconsequenttime-dependenceofthequadratureisthecontent ofRichardson'sTheorem.Intheplane,thequadraturedomaincanbereconstructed fromitsquadratureformula,andquadraturedomainsaredensewithinnaturalclasses ofJordancurves;thesmoothertheclass,thestrongerthetopologyinwhichtheyare densesee[16]andthereferencestherein. Theorem2.5.1Richardson If t isaLaplaciangrowthwith m sinkslocatedat 39

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x i withrates Q i ,thenforanyharmonicfunction u d dt Z t udV = )]TJ/F28 7.9701 Tf 16.18 14.944 Td [(m X i =1 Q i u x i Iftheinitialdomainisnotaquadraturedomain,thentheconnectionofTheorem2.2.2toRichardson'sTheoremrequiresdeningquadraturedomains inthewide sense ,allowingthequadratureformulatoconsistofadistributionwithcompactsupportcontainedinsee[62]and[96].Forsuchgeneralizedquadraturedomains,a distributionwithminimalsupportiscalledamotherbody"forthedomain.The singularitysetoftheSchwarzpotentialgivesasupportingsetforthemotherbody". WorkofGustafssonandSakaiguaranteesexistenceofaquadraturedomain in R n satisfyingaprescribedquadratureformula,butbesidesthespecialexamples in R 4 theonlyexplicitexamplefor n> 2isasphere.Moveover,littlequalitative informationisknownaboutquadraturedomainsinhigherdimensionsbesidesthat theboundaryisanalytic.Forinstance,itisnotevenknownwhetherquadrature domainsaregenerallyalgebraicintheplane,itfollowsfromconditioniv.We makethefollowingconjecture,wherewemeanquadraturedomain"intheclassical, restrictedsenseotherwisethestatementistrivial,sinceanyanalytic,non-singular surfaceisaquadraturedomaininthewidesense: Conjecture2.5.2 Indimensionsgreaterthantwo,thereexistquadraturedomains thatarenotalgebraic. Forthethree-dimensionalexamplefromTheorem2.4.3,wewereabletoisolate thesingularitiesfortheSchwarzpotentialtoasegmentinside.Thus,isaquadrature domaininthewidesenseandhasamotherbodysupportedonthissegment.Weapproximatethedistributionusinganitenumberofpointsonthissegment.Choosing thepoints x k = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1+ k= 2, k =0 ; 1 ;::; 4,wenumericallyintegrate20harmonicbasis functionswritingthemintermsofLegendrepolynomialsover.Ifweassumea quadratureformulainvolvingpointevalutationsatthepoints x k ; 0 ; 0,thenwehave anoverdeterminedlinearsystemforthecoecientsequationsand5unknowns. 40

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Wetaketwosurfaces,andsolvetheleastsquaresproblemforthecoecientsusing thesame5points.Thenthetwosurfacescanbeapproximatelydescribedasthe boundariesofinitialandnaldomainsdrivenbysinksatthesepoints,wherethe totalamountremovedisgivenbythedecreaseinquadratureweight. Figure2.5:Theproleofasupposedinitial a =1andnal a =2domain.Thedriving mechanismtogeneratethesmallerdomainstartingfromthelargercanbeapproximated bycertainamountsofsuctionattheindicatedpoints. Suppose initial isgivenby a =2seestatementofTheorem2.4.3and nal isgivenby a =1.Thenofthetotalvolumeextracted,accordingtotheapproximate description81%isremovedatthepoints 1 ; 0 ; 0,15%atthepoints 1 = 2 ; 0 ; 0, and4%attheoriginSeeFigure2.5.Theaccuracyofthisdescriptionisreectedin thefactthatthenormoftheerrorvectorforbothleastsquaresproblemsisonthe orderof10 )]TJ/F25 7.9701 Tf 6.587 0 Td [(4 2.6Concludingremarks 1. TheequivalentdenitionsofquadraturedomainslistedinSection2.5indicatethe possiblereformulationsoftheLaplaciangrowthproblemeitherintermsofpotential theoryorintermsofholomorphicPDEs.Thepotentialtheoryapproachhasattracted moreattentionandhascertainadvantagessuchasweakformulationsofLaplacian 41

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growth.WehavefocusedontheholomorphicPDEapproach,andinSection2.4we gaveaglimpseofitsmainadvantage: C n techniques. 2. TheremarksattheendofSection2.2mentionaconsequenceoftheSchwarz potentialconjectureregardingLaplaciangrowth.Itwouldbeinterestingifonecould obtainapartialresultintheotherdirectionalongthelinesofSurfacessatisfyingthe SPconjecturearepreservedbyLaplaciangrowth".Thiswouldonlybeinterestingin higherdimensions,sincetheconjectureisalreadyknowntobetrueintheplane. 3. InSection2.2,thediscussioncenteredaroundthecasewhen = =1is constant.Itisnaturaltoconsiderwhen isaxednon-constantentirefunction, andaskifthesolution q todiv r q =1generalizesthedata 1 2 jj x jj 2 intheSchwarz potentialconjecture.WemakethefollowingellipticSchwarzpotentialconjecture". Conjecture2.6.1 Suppose > 0 isentireandthat u solvestheCauchyproblemona nonsingularanalyticsurfacefordiv r u =0 withentiredata.Thenthesingularity setof u iscontainedinthesingularitysetof v ,thesolutionoftheCauchyproblem withdata q ,where q isasolutionofdiv r q =1 Onemightobjecttogeneralizingunresolvedconjectures.Weshouldpointoutthat theSchwarzpotentialconjectureistrueintheplaneandsimpletoprove,whereaswe donotknowifConjecture2.6.1istrueintheplane.OnepieceofevidencefortheSP conjectureisthattheSchwarzpotentialdevelopessingularitiesateverycharacteristic pointoftheinitialsurface[56,Proposition11.3].Asimilarproofshowsthatthisis alsotruefor v ,where f =0 g beingcharacteristicfortheellipticoperatormeans r r + r r =0. 4. AttheendofSection2.3wehavementionedthefactthatinjectionisindependentoftheorderofworkofsourcesandsinks".Inotherwords,theLaplacian growthsdrivenbydierentsourcesandsinkscommute"witheachother.Wecaneven consider,sayhypothetically,injectionateachofinnitelymanyinteriorpointsofa domain.Thenwehaveinnitelymanyprocessesthatcommutewitheachother.This, andespeciallyitsinnitesimalversionwhichfollowsfromtheHadamardvariational 42

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formula,hastheformofanintegrablehierarchy".Tousethepreferredlanguage inthissetting,wehaveacommutingsetofowswithrespecttoinnitelymany generalizedtimes"thetimes"aretheamountsthathavebeeninjectedintoeachof innitelymanysources.Thisholdsinarbitrarydimensionsbuthasrecentlyattracted attentionintwodimensionswhereitisdirectlyconnectedtocertainintegrablehierarchiesinsolitontheorysee[81],[70]and[103].Aspectsofthehigher-dimensionalcase andpossibleconnectionstootherintegrablesystemsseemcompletelyunexplored. 5. Quadraturedomainshavealsoappeared,oftenonlyimplicitly,insolutionsof Euler'sequations.Physically,thisareaofuiddynamicsismuchdierent,involving inviscidowwithvorticity.D.Crowdyhasgivenasurvey[23]ofhisownworkand others'mainlyinthetwo-dimensionalcasewherequadraturedomainshavebeen appliedtovortexdynamics. Theellipsoidisanexampleofaquadraturedomaininthewidesenseforwhich themotherbodyhasbeencalculatedsee[62,Ch.5].Theexteriorgravitational potentialofauniformellipsoidcoincideswiththatofanon-uniformdensitysupported onthetwo-dimensionalfocalellipse"oftheellipsoid.ThisfactwasusedbyDritschel etal[25]asamainstepindevelopingamodelforinteractionofquasi-geostrophic" meteorologicalvortices.Actually,theydidn'tusetheexactdensityofthemother body,butonlythelocationofitssupportinordertochooseasmallnumberof pointvorticesthatgenerateavelocityeldapproximatingthatofanellipsoidof uniformvorticity.Determiningthestrengthoftheapproximatingpointvorticesis nothingmorethaninterpolatingthequadratureformula.Ourcalculationattheend ofSection2.5,andsimilarcalculations,couldhavepromiseforextendingthemodel in[25]toexamplesofnon-ellipsoidalvortices.Animportantmissingingredienthere isastabilityanalysis,whichhasbeencarriedoutforellipsoids. 6. OurintuitionforConjecture2.5.2isbasedontwosuspicionsregardingtheaxiallysymmetriccase.Accordingtothesingularitiesofthefourdimensionalrotationof alimaconconsideredinSection2.3,thequadratureformulainvolvespointevaluations uptoasecond-orderpartialderivative.OnthebasisofL.Karp'sproceduredescribed 43

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inSection2.3,itseemsthatanaxially-symmetricexampleinvolvingonlyapoint evaluationofthefunctionandarst-orderpartialwithrespectto x willhavetobe generatedbyacurvewhoseSchwarzfunctionhasanessentialsingularityattheorigin. Then,theconformalmapwouldbetranscendental.In R 3 weexpectthesituation tobeatleastasbad.Following[44,Ch.s4and5],onecanwriteanintegralformula involvingaGausshypergeometricfunctionforthesolutionofaCauchyproblemforan n -dimensionalaxially-symmetricpotential.Thethreedimensionalcaseoftheformula hasthesameformasthefour-dimensionalcase,excepttheinvolvedhypergeometric functionistranscendentalinsteadofrational. 44

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3AlgebraicDirichletProblems 3.1TheSearchforSingularitiesofSolutionstotheDirichletProblem: RecentDevelopments Thissectionistakenfromthesurveyarticle[57]writtenjointlywithDmitryKhavinsonandbasedonaninvitedtalkdeliveredbyDmitryKhavinsonattheCRMworkshoponHilbertSpacesofAnalyticFunctionsheldatCRM,UniversitedeMontreal, December8-12,2008. 3.1.1Themainquestion Letbeasmoothlyboundeddomainin R n .ConsidertheDirichletProblemDP inofndingthefunction u ,say, 2 C 2 T C andsatisfying 8 < : u =0 u j )]TJ/F19 11.9552 Tf 9.108 1.793 Td [(= v ; .1.1 where= P n j =1 @ 2 @x 2 j and)-363(:= @ ;v 2 C \051.Itiswellknownsincetheearly20th centuryfromworksofPoincare,C.Neumann,Hilbert,andFredholmthatthesolution u existsandisunique.Also,since u isharmonicin,hencereal-analyticthere,no singularitiescanappearin.Moreover,assuming)-278(:= @ toconsistofreal-analytic hypersurfaces,themorerecentanddicultresultsonellipticregularity"assureus thatifthedata v isreal-analyticinaneighborhoodof then u extendsasarealanalyticfunctionacross @ intoanopenneighborhood 0 of .Intwodimensions, thiscanbedoneusingthereectionprinciple.Inhigherdimensions,theboundary 45

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canbebiholomorphicallyattened",butthisleadstoageneralellipticoperatorfor whichthereectionprincipledoesnotapply.Instead,analyticitymustbeshownby directlyverifyingconvergenceofthepowerseriesrepresentingthesolutionthrough dicultestimatesonthederivativessee[41]. Question Supposethedata v isarestrictionto)-330(ofaverygood"function,sayan entirefunctionofvariables x 1 ;x 2 ;:::;x n .Inotherwords,thedatapresentsnoreasons whatsoeverforthesolution u of.1.1todevelopsingularities. iCanwethenassertthatallsolutions u of.1.1withentiredata v x arealso entire? iiIfsingularitiesdooccur,theymustbecausedbygeometryof)-347(interactingwith thedierentialoperator.Canwethennddata v 0 thatwouldforcetheworst possiblescenariotooccur?Moreprecisely,foranyentiredata v ,thesetofpossible singularitiesofthesolution u of.1.1isasubsetofthesingularitysetof u 0 ,the solutionof.1.1withdata v 0 3.1.2TheCauchyProblem AninspirationtothisprogramlaunchedbyH.S.ShapiroandD.Khavinsonin[64] comesfromreasonablesuccesswithasimilarprograminthemid1980'sregarding theanalyticCauchyProblemCPforellipticoperators,inparticular,theLaplace operator.Forthelatter,weareseekingafunction u with u =0near)-237(andsatisfying theinitialconditions 8 < : u )]TJ/F27 11.9552 Tf 11.955 0 Td [(v j )]TJ/F19 11.9552 Tf 9.107 1.793 Td [(=0 r u )]TJ/F27 11.9552 Tf 11.955 0 Td [(v j )]TJ/F19 11.9552 Tf 9.108 1.793 Td [(=0 ; .1.2 where v isassumedtobereal-analyticinaneighborhoodof.Supposeasbefore thatthedata v isagood"functione.g.apolynomialoranentirefunction.In thatcontext,thetechniquesdevelopedbyJ.Leray[73]inthe1950'sandjointly withL.GardingandT.Kotake[42]togetherwiththeworksofP.Ebenfelt[33],G. Johnsson[52],and,independently,byB.SterninandV.Shatalov[94]inRussiaand 46

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theirschoolproducedamoreorlesssatisfactoryunderstandingofthesituation.To mentionbriey,theanswerfortheCPtoquestioniintwodimensionsisessentially never"unless)-395(isalinewhileforiithedataminingallpossiblesingularitiesof solutionstotheCPwithentiredatais v 0 = j x j 2 = P x 2 j see[62],[55],[93],and[56] andreferencestherein. 3.1.3TheDirichletproblem:Whendoesentiredataimplyentiresolution? LetusraiseQuestioniagainfortheDirichletProblem:Doesrealentiredata v implyentiresolution u of.1.1? Inthissectionandthenext, P willdenotethespaceofpolynomialsand P N thespaceofpolynomialsofdegree N .Thefollowingprettyfactgoesbacktothe 19thcenturyandcanbeassociatedwiththenamesofE.Heine,G.Lame,M.Ferrers, andprobablymanyotherscf.[56].Theproofisfrom[64]cf.[10],[12]. Proposition3.1.1 If := f x : P x 2 j a 2 j )]TJ/F19 11.9552 Tf 10.931 0 Td [(1 < 0 ;a 1 >:::>a n > 0 g isanellipsoid,then anyDPwithapolynomialdataofdegree N hasapolynomialsolutionofdegree N Proof. Let q x = P x 2 j a 2 j )]TJ/F19 11.9552 Tf 12.038 0 Td [(1bethedeningfunctionfor)-295(:= @ .Thelinearmap T : P qP sendsthenite-dimensionalspace P N intoitself. T isinjectiveby themaximumprincipleand,therefore,surjective.Hence,forany P degP 2we cannd P 0 ,deg P 0 deg P )]TJ/F19 11.9552 Tf 12.415 0 Td [(2. TP 0 = qP 0 = P u = P )]TJ/F27 11.9552 Tf 12.415 0 Td [(qP 0 isthenthe desiredsolution. Thefollowingresultwasprovedin[64]. Theorem3.1.2 AnysolutiontoDP.1.1inanellipsoid withentiredataisalso entire. Lateron,D.Armitagesharpenedtheresultbyshowingthattheorderand thetypeofthedataarecarriedover,moreorless,tothesolution[7].Thefollowing conjecturehasalsobeenformulatedin[64]. 47

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Conjecture3.1.3 Ellipsoidsaretheonlyboundeddomainsin R n forwhichTheorem 3.1.2holds,i.e.,ellipsoidsaretheonlydomainsinwhichentiredataimpliesentire solutionfortheDP.1.1. In2005,H.Render[86]provedthisconjectureforall algebraically boundeddomainsdenedasboundedcomponentsof f x < 0 ; 2 P N g suchthat f x =0 g isa bounded setin R n or,equivalently,theseniorhomogeneouspart N x of iselliptic,i.e., j N x j C j x j N forsomeconstant C .For n =2,aneasierversion ofthisresultwassettledin2001byM.ChamberlandandD.Siegel[22].Belowwe outlinetheirargument,whichestablishessimilarresultsasRender'sforthefollowing modiedconjecture. Conjecture3.1.4 Ellipsoidsaretheonlysurfacesforwhichpolynomialdataimplies polynomialsolution. Remark: WewillreturntoRender'sTheorembelow.Fornowletusnotethat, unfortunately,italreadytellsusnothingevenin2dimensionsformanyperturbations ofaunitdisk,e.g.,:= f x 2 R 2 : x 2 + y 2 )]TJ/F19 11.9552 Tf 12.548 0 Td [(1+ "h x;y < 0 g where,say, h isa harmonicpolynomialofdegree > 2. 3.1.4Whendoespolynomialdataimplypolynomialsolution? Let = f x =0 g beabounded,irreduciblealgebraiccurvein R 2 .IftheDPposed on haspolynomialsolutionwheneverthedataisapolynomial,thenasChamberland andSiegelobserved,a isanellipseorbthereexistsdata f 2 P suchthatthe solution u 2 P ofDPhasdeg u> deg f Incaseb u )]TJ/F27 11.9552 Tf 12.312 0 Td [(f j =0impliesthat divides u )]TJ/F27 11.9552 Tf 12.312 0 Td [(f byHilbert'sNullstelensatz,and,sincedeg u = M> deg f u M = k g l where k and u M arethesenior homogeneoustermsof and u respectively.Theseniortermof u musthavetheform u M = az M + b z M since u M isharmonic.Hence, u M factorsintolinearfactorsandso must k .Hence isunbounded.Thisgivesthefollowingresult[22]. 48

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Theorem3.1.5 Supposedeg > 2 and issquare-free.IftheDirichletproblem posedon f =0 g hasapolynomialsolutionforeachpolynomialdata,thenthesenior partof ,whichwedenoteby N ,oforder N ,factorsintoreallinearterms,namely, N = n Y j =0 a j x )]TJ/F27 11.9552 Tf 11.955 0 Td [(b j y ; where a j b j aresomerealconstantsandtheanglesbetweenthelines a j x )]TJ/F27 11.9552 Tf 12.177 0 Td [(b j y =0 forall j ,arerationalmultiplesof ThistheoremsettlesConjecture3.1.4forboundeddomains f x < 0 g suchthattheset f x =0 g isboundedin R 2 .However,thetheoremleavesopen simplecasessuchas x 2 + y 2 )]TJ/F19 11.9552 Tf 11.956 0 Td [(1+ x 3 )]TJ/F19 11.9552 Tf 11.955 0 Td [(3 xy 2 Example: Thecurve y y )]TJ/F27 11.9552 Tf 12.025 0 Td [(x y + x )]TJ/F27 11.9552 Tf 12.026 0 Td [(x =0seegure3.1satisesthenecessary conditionimposedbythetheorem.Moreover,anyquadraticdatacanbematchedon itbyaharmonicpolynomial.Forinstance, u = xy y 2 )]TJ/F27 11.9552 Tf 12.164 0 Td [(x 2 solvestheinterpolation problemitismisleadingtosayDirichlet"problem,sincethereisnoboundedcomponentwithdata v x;y = x 2 .Ontheotherhand,onecanshownon-triviallythat thedata x 3 doesnothavepolynomialsolution. 3.1.5Dirichlet'sProblemandOrthogonalPolynomials Mostrecently,D.KhavinsonandN.Stylianopoulosshowedthatifforapolynomial datatherealwaysexistsapolynomialsolutionoftheDP.1.1,withanadditional constraintonthedegreeofthesolutionintermsofthedegreeofthedataseebelow, thenisanellipse[65].Thisresultdrawsonthe2007paperofM.PutinarandN. Stylianopoulos[85]thatfoundasimplebutsurprisingconnectionbetweenConjecture 3.1.4in R 2 andBergmanorthogonalpolynomials,i.e.polynomialsorthogonalwith respecttotheinnerproduct h p;q i := R p qdA ,where dA istheareameasure.To understandthisconnectionletusconsiderthefollowingproperties: 1.Thereexists k suchthatforapolynomialdataofdegree n therealwaysexistsa polynomialsolutionoftheDP.1.1posedonofdegree n + k 49

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Figure3.1:Acubiconwhichanyquadraticdatacanbematchedbyaharmonicpolynomial. 2.Thereexists N suchthatforall m;n ,thesolutionof.1.1withdata z m z n isa harmonicpolynomialofdegree N )]TJ/F19 11.9552 Tf 10.066 0 Td [(1 m + n in z andofdegree N )]TJ/F19 11.9552 Tf 10.066 0 Td [(1 n + m in z 3.Thereexists N suchthatorthogonalpolynomials f p n g ofdegree n onsatisfy anite N +1-recurrencerelation,i.e. zp n = a n +1 ;n p n +1 + a n;n p n + ::: + a n )]TJ/F28 7.9701 Tf 6.586 0 Td [(N +1 p n )]TJ/F28 7.9701 Tf 6.586 0 Td [(N +1 ; where a n )]TJ/F28 7.9701 Tf 6.587 0 Td [(j;n areconstantsdependingon n 4.TheBergmanorthogonalpolynomialsofsatisfyanite-termrecurrencerelation,i.e.,foreveryxed k> 0,thereexistsan N k > 0,suchthat a k;n = h zp n ;p k i =0, n N k 5.Conjecture3.1.4holdsfor. PutinarandStylianopoulosnoticedthatwiththeadditionalminorassumption thatpolynomialsaredensein L 2 a ,propertiesandareequivalent.Thus, 50

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theyobtainedasacorollarybywayofTheorem3.1.5fromtheprevioussection thattheonlyboundedalgebraicsetssatisfyingpropertyareellipses.Wealso have , ,and .KhavinsonandStylianopoulosusedthe equivalenceofpropertiesandtoprovethefollowingtheoremwhichhasan immediatecorollary. Theorem3.1.6 Suppose @ is C 2 -smooth,andorthogonalpolynomialson satisfy anite N +1 -recurrencerelation,inotherwordsproperty issatised.Then, N =2 and isanellipse. Corollary3.1.7 Suppose @ isa C 2 -smoothdomainforwhichthereexists N such thatforall m;n ,thesolutionof.1.1withdata z m z n isaharmonicpolynomialof degree N )]TJ/F19 11.9552 Tf 11.818 0 Td [(1 m + n in z andofdegree N )]TJ/F19 11.9552 Tf 11.818 0 Td [(1 n + m in z .Then N =2 and isanellipse. Proof. [Sketchofproof]First,onenotesthatallthecoecientsintherecurrence relationarebounded.Dividebothsidesoftherecurrencerelationaboveby p n and takethelimitofanappropriatesubsequenceas n !1 .Knownresultsonasymptotics oforthogonalpolynomialssee[101]givelim n !1 p n +1 p n = z oncompactsubsetsof C n ,where z istheconformalmapoftheexterioroftotheexteriorofthe unitdisc.Thisleadstoa nite Laurentexpansionat 1 for w = )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 w .Thus, w isarationalfunction,so ~ := C n isanunboundedquadraturedomain,and theSchwarzfunctioncf.[24],[97]of @ S z = z on @ hasameromorphic extensionto ~ .Suppose,forthesakeofbrevityandtoxtheideas,forexample, that S z = cz d + P M j =1 c j z )]TJ/F28 7.9701 Tf 6.587 0 Td [(z j + f z ,where f 2 H 1 ~ ,and z j 2 ~ .Sinceour hypothesisisequivalenttosatisfyingpropertydiscussedabove,thedata zP z = z Q n j =1 z )]TJ/F27 11.9552 Tf 11.955 0 Td [(z j haspolynomialsolution, g z + h z totheDP.On)-330(wecanreplace z with S z .Write h z = h # z ,where h # isapolynomialwhosecoecientsare complexconjugatesoftheircounterpartsin h .Wehaveon)]TJ/F27 11.9552 Tf -90.552 -37.36 Td [(S z P z = g z + h # S z ; .1.3 51

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whichisactuallytrueo)-300(sincebothsidesoftheequationareanalytic.Near z j ,the left-hand-sideofthisequationtendstoanitelimitsince S z P z isanalyticin ~ n1 !whiletheright-hand-sidetendsto 1 unlessthecoecient c j iszero.Thus, S z = cz d + f z : .1.4 Usingpropertyagainwithdata j z j 2 = z z wecaninferthat d =1.Hence, ~ isa nullquadraturedomain.Sakai'stheorem[91]impliesnowthatisanellipse. Remark: Itiswell-knownthatfamiliesoforthogonalpolynomialsonthelinesatisfy a3-termrecurrencerelation.P.Durenin1965[26]alreadynotedthatin C theonly domainswithreal-analyticboundariesinwhichpolynomialsorthogonalwithrespect toarc-lengthontheboundarysatisfy3-termrecurrencerelationsareellipses.L. Lempert[72]constructedpeculiarexamplesof C 1 non-algebraicJordandomainsin whichnoniterecurrencerelationforBergmanpolynomialsholds.Theorem3.1.6 showsthatactuallythisistruefor all C 2 -smoothdomainsexceptellipses. 3.1.6LookingforsingularitiesofthesolutionstotheDirichletProblem Onceagain,inspiredbyknownresultsinthesimilarquestforsolutionstotheCauchy problem,onecouldexpect,e.g.,thatthesolutionstotheDP.1.1exhibitbehavior similartothoseoftheCP.1.2.Inparticular,itseemednaturaltosuggestthatthe singularitiesofthesolutionstotheDPoutsidearesomehowassociatedwiththe singularitiesoftheSchwarzpotentialfunctionof @ whichdoesindeedcompletely determine @ cf.[62],[97].ItturnedoutthatsingularitiesofsolutionsoftheDPare muchmorecomplicatedthanthoseoftheCP.Alreadyin1992inhisthesis,P.Ebenfelt showed[30]thatthesolutionofthefollowinginnocent"DPin:= f x 4 + y 4 )]TJ/F19 11.9552 Tf 10.502 0 Td [(1 < 0 g theTV-screen" 8 < : u =0 u j @ = x 2 + y 2 .1.5 52

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Figure3.2:AplotoftheTVscreen" f x 4 + y 4 =1 g alongwiththersteightsingularities plottedascirclesencounteredbyanalyticcontinuationofthesolutiontoDP.1.5. hasaninnitediscretesetofsingularitiesofcourse,symmetricwithrespectto90 rotationsittingonthecoordinateaxesandrunningto 1 seegure3.2. ToseethedierencebetweenanalyticcontinuationofsolutionstoCPandDP, notethatfortheformer @u @z j := @ = v z z; z = v z z;S z ; .1.6 andsince @u @z isanalytic,.1.6allows u z tobecontinuedeverywheretogetherwith v and S z ,theSchwarzfunctionof @ .FortheDPwehaveon)]TJ/F27 11.9552 Tf -8.105 -45.828 Td [(u z; z = v z; z .1.7 53

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for u = f + g where f and g areanalyticin.Hence,.1.7becomes f z + g S z = v z;S z : .1.8 Now, v z;S z doesindeedforentire v extendtoanydomainfreeofsingularitiesof S z ,but.1.8,evenwhen v isreal-valuedsothat g = f ,presentsa non-trivialfunctionalequationsupportedbyarathermysteriouspieceofinformation that f isanalyticin..1.8howevergivesaninsightastohowtocapturethe DP-solution'ssingularitiesbyconsideringtheDPaspartofaGoursatproblemin C 2 or C n ingeneral.ThelatterGoursatproblemcanbeposedasfollowscf.[95]. Givenacomplex-analyticvariety ^ )-478(in C n ^ )]TJ/F32 11.9552 Tf 9.307 8.966 Td [(T R n =)-535(:= @ ,nd u : P n j =1 @ 2 @z 2 j u =0near ^ )-438(andalsoin R n sothat u j ^ )]TJ/F19 11.9552 Tf 11.371 3.431 Td [(= v ,where v is,say,an entirefunctionof n complexvariables.Thus,if ^ )-414(:= f z =0 g ,where is,say, anirreduciblepolynomial,wecan,e.g.,ponderthefollowingextensionofConjecture 3.1.3: Question: Forwhichpolynomials caneveryentirefunction v besplitFischer decompositionas v = u + h ,where u =0and u h areentirefunctionscf.[39], [95]? 3.1.7Render'sbreakthrough TryingtoestablishConjecture3.1.3,H.Render[86]madethefollowingingenious step.Heintroducedthe real versionoftheFischerspacenorm h f;g i = Z R n f ge j x j 2 dx; .1.9 where f and g arepolynomials.Originally,theFischernormintroducedbyE.Fischer [39]requirestheintegrationtobecarriedoverallof C n andhasthepropertythat multiplicationbymonomialsisadjointtodierentiationwiththecorrespondingmultiindexe.g.,multiplicationby P n j =1 x 2 j isadjointtothedierentialoperator.This 54

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propertyisonlypartiallypreservedfortherealFischernorm.Moreprecisely[86], h f;g i = h f; g i +2 deg f )]TJ/F27 11.9552 Tf 11.956 0 Td [(deg g h f;g i .1.10 forhomogeneous f g Suppose u solvestheDPwithdata j x j 2 on @ f P =0: deg P =2 k;k> 1 g .Then u )-326(j x j 2 = Pq foranalytic q ,andthus k Pq =0.Using.1.10, thisnon-triviallyimpliesthattherealFischerproduct h Pq m +2 k ;q m i betweenall homogeneouspartsofdegree m +2 k and m of Pq and q ,respectively,iszero.By atourdeforceargument,Renderusedthisalongwithanaddedassumptiononthe seniortermof P seebelowtoobtainestimates frombelow forthedecayofthe normsofhomogeneouspartsof q .This,inturnyieldsanif-and-only-ifcriterionfor convergenceintherealballofradius R oftheseriesforthesolution u = P 1 m =0 u m u m homogeneousofdegree m .LetusstateRender'smaintheorem. Theorem3.1.8 Let P beanirreduciblepolynomialofdegree 2 k k> 1 .Suppose P iselliptic,i.e.theseniorterm P 2 k of P satises P 2 k x c P j x j 2 k ,forsomeconstant c P .Let berealanalyticin fj x j 2,andthefollowingFischerdecomposition holds: j x j 2 = P + u u =0.Hence, k P =0and cannotbeanalytically continuedbeyonda nite ballofradius R = C P < 1 ,acontradiction. 55

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Caution: Wewanttostressagainthat,unfortunately,thetheoremstilltellsus nothingforsaysmallperturbationsofthecirclebyanon-elliptictermofdegree 3, e.g., x 2 + y 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1+ x 3 )]TJ/F19 11.9552 Tf 11.955 0 Td [(3 xy 2 3.1.8Backto R 2 :lightningbolts Returntothe R 2 settingandconsiderasbeforeourboundary @ ofadomainas partofanintersectionofananalyticRiemannsurface ^ )-374(in C 2 with R 2 .Roughly speaking,ifsay @ isasubsetofthealgebraiccurve)-277(:= f x;y : x;y =0 g ,where isanirreduciblepolynomial,then ^ )-402(= f X;Y 2 C 2 : X;Y =0 g .Nowlook attheDirichletproblemagaininthecontextoftheGoursatproblem:Given,say,a polynomialdata P ,nd f g holomorphicfunctionsofonevariablenear ^ )-303(apieceof ^ )-326(containing @ ^ )]TJ/F32 11.9552 Tf 9.306 8.966 Td [(T R 2 suchthat u = f z + g w j ^ )]TJ/F19 11.9552 Tf 9.107 3.432 Td [(= P z;w ; .1.11 wherewehavemadethelinearchangeofvariables z = X + iY w = X )]TJ/F27 11.9552 Tf 12.661 0 Td [(iY so w = z on R 2 = f X;Y : X;Y arebothreal g .Obviously, u =4 @ 2 @z@w =0and u matches P on @ .Thus,theDPin R 2 hasbecomeaninterpolationproblem in C 2 ofmatchingapolynomialonanalgebraicvarietybyasumofholomorphic functionsineachvariableseparately.Supposethatforallpolynomials P thesolutions u of.1.11extendasanalyticfunctionstoaball B = fj z j 2 + j w j 2
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wasindependentlyobservedbyE.Study[100],H.Lewy[74],andL.HansenandH. S.Shapiro[46].Indeed,assignalternatingvalues 1forthemeasuresupportedat thefourpoints p 0 := z 1 ;w 1 q 0 := z 1 ;w 2 p 1 := z 2 ;w 2 ,and q 1 := z 2 ;w 1 .Then R f + g d = f z 1 + g w 1 )]TJ/F27 11.9552 Tf 12.638 0 Td [(f z 1 )]TJ/F27 11.9552 Tf 12.638 0 Td [(g w 2 + f z 2 + g w 2 )]TJ/F27 11.9552 Tf 12.638 0 Td [(f z 2 )]TJ/F27 11.9552 Tf 12.639 0 Td [(g w 1 =0 forallholomorphicfunctions f and g ofonevariable.Thisisanexampleofaclosed lightningboltLBwithfourvertices.Clearly,theideacanbeextendedtoanyeven numberofvertices. Denition3.1.10 AcomplexclosedlightningboltLBoflength2 n +1isanite setofpointsvertices p 0 ;q 0 ;p 1 ;q 1 ;:::;p n ;q n ;p n +1 ;q n +1 suchthat p 0 = p n +1 ,andeach complexlineconnecting p j to q j or q j to p j +1 haseither z or w coordinatexedand theyalternate,i.e.,ifwearrivedat p j with w coordinatexedthenwefollowto q j with z xedetc. Forreal"domainslightningboltswereintroducedbyArnoldandKolmogorov inthe1950stostudyHilbert's13thproblemsee[67]andthereferencestherein. Thefollowingtheoremhasbeenprovedin[14]seealso[15]. Theorem3.1.11 Let beaboundedsimplyconnecteddomainin C = R 2 suchthat theRiemannmap : D = fj z j < 1 g isalgebraic.Thenallsolutionsofthe DPwithpolynomialdatahaveonlyalgebraicsingularitieswhichoccuronlyatbranch pointsof withthebranchingorderoftheformerdividingthebranchingorderofthe latteri )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 isarationalfunction.Thisinturnisknowntobeequivalentto being aquadraturedomain. Proof. [Ideaofproof:]Thehypothesesimplythatthesolution u = f + g extends asasingle-valuedmeromorphicfunctionintoa C 2 -neighborhoodof ^ .Byanother theoremof[14],onecanndunless )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 isrationalacontinuousfamilyofclosedLBs on ^ )-313(ofboundedlengthavoidingthepolesof u .Hence,themeasurewithalternating values 1ontheverticesofanyoftheseLBsannihilatesallsolutions u = f z + 57

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Figure3.3:AMapleplotofthecubic8 x x 2 )]TJ/F49 10.9091 Tf 8.864 0 Td [(y 2 +57 x 2 +77 y 2 )]TJ/F15 10.9091 Tf 8.864 0 Td [(49=0,showingthebounded componentandoneunboundedcomponenttherearetwootherunboundedcomponents furtheraway. g w holomorphicon ^ ,butdoesnot,ofcourse,annihilateallpolynomialsof z w Therefore, )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 mustberational,i.e.isaquadraturedomain[95]. Theauthorofthisthesis[77]orseeSection3.2belowhasrecentlyconstructed someotherexamplesofLBsoncomplexiedboundariesofplanardomainswhichdo notsatisfythehypothesisofRender'stheorem.TheLBsvalidateConjecture3.1.3and produceanestimateregardinghowfarintothecomplement C n thesingularitiesmay develop.Forinstance,thecomplexicationofthecubic,8 x x 2 )]TJ/F27 11.9552 Tf 9.386 0 Td [(y 2 +57 x 2 +77 y 2 )]TJ/F19 11.9552 Tf 9.386 0 Td [(49= 0hasalightningboltwithsixverticesinthenon-physicalplanewhere z and w are real,i.e., x isrealand y isimaginaryseegure3.3foraplotofthecubicintheplane where x and y arerealandseegure3.4forthenon-physical"sliceincludingthe lightningbolt.Ifthesolutionwithappropriatecubicdataisanalyticallycontinued inthedirectionoftheclosestunboundedcomponentofthecurvedening @ ,it willhavetodevelopasingularitybeforeitcanbeforcedtomatchthedataonthat component. 58

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Figure3.4:Alightningboltwithsixverticesonthecubic2 z + w z 2 + w 2 +67 zw )]TJ/F15 10.9091 Tf 9.881 0 Td [(5 z 2 + w 2 =49inthenon-physicalplanewith z and w real,i.e. x realand y imaginary. 59

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3.1.9Furtherquestions Intwodimensionsoneofthemainresultsin[14]yieldsthatdisksaretheonlydomains forwhichallsolutionsoftheDPwithrationalin x;y data v arerational.Thefact thatinadiskeveryDPwithrationaldatahasarationalsolutionwasobservedina seniorthesisofT.FergussonatU.ofRichmond[90].Ontheotherhand,algebraic datamayleadtoatranscendentalsolutionevenindiskssee[32],alsocf.[34].In dimensions3andhigher,rationaldataonthespheree.g., v = 1 x 1 )]TJ/F28 7.9701 Tf 6.586 0 Td [(a j a j > 1yields transcendentalsolutionsof.1.1,althoughwehavenotbeenabletoestimatethe locationofsingularitiespreciselycf.[32]. Itisstillnotclearonanintuitivelevelwhyellipsoidsplaysuchadistinguished roleinprovidingexcellent"solutionstoDPwithexcellent"data.Averysimilar question,importantforapplications,whichactuallyinspiredtheprogramlaunched in[64]onsingularitiesofthesolutionstotheDPgoesbacktoRaleighandconcerns singularitiesofsolutionsoftheHelmholtzequation[ )]TJ/F27 11.9552 Tf 12.259 0 Td [( 2 ] u =0, 2 R instead. Theminussignwillguaranteethatthemaximumprincipleholdsand,consequently, ensuresuniquenessofsolutionsoftheDP.Tothebestofourknowledge,thistopic remainsvirtuallyunexplored. 3.2Dirichlet'sProblemandComplexLightningBolts Thissectionistakenfromthepaper[77]publishedinthejournalComputational MethodsandFunctionTheory".Weinvestigatesomeofthetopicssurveyedinthe previoussection. WeconsidertheDirichletproblemintheplanewithentiredataonalgebraic curves.Morespecicallywewillbeinterestedinwheresingularitiesdevelopwhena solutioniscontinuedanalytically.Ourapproachinvolvesnitelysupportedannihilatingmeasuressupportedonnitesetscalledlightningbolts.Lightningboltswere rstusedintherealsettingbyKolmogorovandArnoldtosolveHilbert's13thproblem.After[77]waspublished,H.Renderpointedoutthattheexpositoryexample 60

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involvingthewaveequationistreatedmoregenerallyinthepaperofF.John[51]. 3.2.1AlgebraicallyposedDirichletproblems ConsidertheDirichletproblemposedinsideaconnectedcomponentofanalgebraic curvewithreal-entiredata.Wecanextendthesolutiontoatleastaneighborhood byreection"usingthecurve'sSchwarzfunctionsee[97].Inthissection,wetreat thequestionofhowfarfromthecurveofinitialdatawecananalyticallycontinuethe solution. Thisquestionwasinvestigatedin[30]whereEbenfeltdescribedthesetof singularitiesdevelopedbysolutionswithquadraticdataonthecurve x 4 + y 4 =1.In abroadersetting,hemadetherststepinconrmingtheconjectureofKhavinson andShapiro[64]thattheellipseistheonlycurveforwhichallsolutionswithentire dataareentire.RecentlyRenderconrmedthisconjectureinalldimensionsfora largeclassofeven-degreevarieties[86].Hismethodreliesondicultestimates frombelowontheellipticoperatoractingonhomogeneouspartstoboundtheradius ofconvergence.Anupperboundforthemaximumdiscofconvergencegivesanupper boundforthemaximumdiscofanalyticity. Wepursueherethetechniqueusedin[15]toshowfailureofanalyticityof solutions.WecombinethisapproachwiththeconceptoftheVekuahulltodevelop amethodtolocateatleastinprinciplesingularities.Thenwegiveasimpleproof oftheproximityofsingularitiestotheinitialcurveforEbenfelt'sexample,andin nallywediscussexamplesnotcoveredbyRender'sapproach.First,weintroduce theideasintherealsettingwherewewillstrayslightlyfromourmaininterestin ordertoillustratethemethod. 3.2.2RealLightningBoltsandIll-posedProblemsfortheWaveEquation Recallthedenition,givenintheprevioussection,ofalightningbolt.Briey,the pointshereafter,verticesinalightningboltaregivenbytheverticesofapolygonal 61

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arcwhosesegmentsareparalleltothecoordinateaxes.Forexample,,1,,4,1,4,-1,1isalightningbolt.Inthisparticularcase,thesequenceobtainedbyaugmentingthislightningboltwithitsrstvertex,,1,isagainalightningbolt.We saythatalightningboltis closed ifitremainsalightningboltwhenaugmentedwith itsrstvertex.Clearly,aclosedlightningboltalwayshasanevennumberofvertices. MeasuresconstructedonlightningboltswereusedbyKolmogorovandArnold tosolveHilbert's13thproblemregardingthesolutionof7thdegreeequationsusing functionsoftwoparameters.Lightningboltsplayedacentralroleindetermining whenafunctionofseveralrealvariablescanberepresentedasasuperpositionof functionsoffewervariables.Thesuperpositionproblemwasfurtherdevelopedin approximationtheory[67].Inthecasewhenafunctionoftwovariablesistobe matchedonaclosedcurvebyasumoffunctionsineachvariableseparately,this problemisrelatedtosolvingaboundaryvalueproblemforthewaveequationinone spatialdimension. Inordertoillustratetheuseofreallightningbolts,wepursuethewaveequation withDirichlet-typedata.Ifthepropagationspeedhasbeennormalizedsothatthe waveoperatorhastheform @ 2 @x 2 )]TJ/F28 7.9701 Tf 15.199 4.707 Td [(@ 2 @t 2 ,thenchangingvariablestothecharacteristic coordinates = x + t; = x )]TJ/F27 11.9552 Tf 12.339 0 Td [(t convertstheoperatorto @ 2 @@ ,andallsolutions, u tothehomogeneouswaveequation @ 2 u @@ =0thentaketheform u ; = f + g see[50]. Thisreformulatestheproblemintooneofmatchingatwo-variablefunctionthe dataontheboundarycurvebyasumoffunctionsineachvariableseparately.This placesusnearlyinthesamesettingasthesuperpositionproblem.Manyresultscarry overimmediately,perhapsinaweakerform.Forinstance,onthetrianglewithvertices ; 0, = 2 ; 0, ; 1whichclearlyhasnoclosedlightningbolts,anycontinuous functioncanbeuniformlyapproximatedbutinsomecasesnotrepresentedbysums ofcontinuousfunctionsineachvariableseparatelyforaproofsee[67].Thus,wecan solve"thewaveequationwithsolutionsthatapproximatelymatchanycontinuous boundarydata. 62

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Necessaryconditionsandnegativeanswersareprovidedbythefollowingsimpletheoremandtheideasinitsproof. Theorem3.2.1 Let Q beasubsetof R 2 .If Q containsaclosedlightningboltoflength 2 n ,thenthereisapolynomial P ; ofdegree n whichcannotbeapproximatedon Q byfunctionsoftheform f + g Proof. Withoutlossofgenerality,suppose 1 = 2 .Let P ; = )]TJ/F27 11.9552 Tf 10.003 0 Td [( 1 n Q j =2 )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 j Then P ; isofdegree n andiszeroateachvertexexceptthesecond.Considerthe measurewhichassignsthevalue )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 k tothe kth vertex, k ; k ,inthelightningbolt. Integrating P ; againstthismeasureproducesthevalue P 2 ; 2 .Integratingany f + g againstthismeasuregives 2 n P k )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 k [ f k + g k ]= 2 n P k )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 k f k + P k )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 k g k .Eachofthesesumstelescopestozero.BytheHahn-BanachTheorem, P ; cannotberepresentedorevenuniformlyapproximatedbysums f + g Example: In[49]aboundaryvalueproblemforthenon-homogeneouswave equationisposedonatrianglewith,using ; -coordinates,vertices ; 0, ; 0, ; 2.Dataisprescribedtobezeroalongthehypotenuseandtosatisfy u t; 0= )]TJ/F27 11.9552 Tf 9.298 0 Td [(u ;t alongthecharacteristicedges.Itisshownthatthisrestrictionissucientto guaranteeexistenceanduniqueness.Forthehomogeneousproblemonthistriangle, theproofofTheorem3.2.1indicatesthatanecessaryconditionforexistenceofsolutionsisthattheboundarydatamustbeannihilatedbyeveryalternatingmeasure constructedonaclosedlightningboltresidingontheboundary.Forthefamilyoflightningbolts t; 0, ; 0, t;t ;t thisconditionis u t; 0 )]TJ/F27 11.9552 Tf 9.626 0 Td [(u ; 0+ u ;t )]TJ/F27 11.9552 Tf 9.626 0 Td [(u t;t =0. Combiningthiswiththerestriction u t; 0= )]TJ/F27 11.9552 Tf 9.299 0 Td [(u ;t impliesthat u ; iszero.This indicateshowuniquenessofsolutionsforthenon-homogeneousequationfollowsfrom therestrictionontheboundaryvalues.Indeed,thedierenceoftwosolutionsfor thenon-homogeneousequationsisasolutionforthehomogeneousequationthatstill satisestherestrictionontheboundary,sothatitmustbeidenticallyzero. 63

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Inthisexample,theclosedlightningboltsarejust characteristicparallelograms see[50].Next,consideranexampleofaclosedlightningboltwithmorethanfour vertices. Placedataontherectanglewith x;t -coordinates ; 0 ; ;b ; a;b ; ;a The ; coordinatesfortheverticesare ; 0 ; b; )]TJ/F27 11.9552 Tf 9.298 0 Td [(b ; a + b;a )]TJ/F27 11.9552 Tf 12.167 0 Td [(b ; a; )]TJ/F27 11.9552 Tf 9.298 0 Td [(a ,which isarectangletilted45 .Studyinglightningboltsonarectanglewiththisparticular tiltbecomesasimpleproblemindynamicalbilliards,sincesuccessiveverticesarethe sameasthewall-collisionsofaparticlewiththesameinitialconditions. Figure3.5:Fortherectanglewithsideratio1:2,thereareinnitelymanylightningbolts withsixvertices.Thehorizontallinerstreturnsafterthreeintersectionsgivingtherst halfofthelightningbolt. Findingaclosedlightningboltcorrespondstondingaperiodicorbitexcept inthecasewhentheorbithitsacorner.Tilingtheplanewithsuccessivereections ofthisrectangleoveritssidesreducesthebookkeepingofaparticle'sorbittothatof notingthecrossingpointsofahorizontallineseegure3.5.Ifthesidelengthsof therectanglehaverationalratio,anyhorizontallinewilleventuallycrosstwotiles" atthesamepointgivingaperiodicorbit,inwhichcasewehaveaclosedlightning bolt.Toseewhy,supposetherectanglehassidesoflength1and q where q isrational. Thenthecrossingpositionsonthesidesoflength1areobtainedbyadding,modulo1, q tothepreviouscrossingposition.Weobtaininnitelymanyclosedlightningbolts ofthesamelength.ByTheorem3.2.1,inordertohaveexistenceofasolutionto thehomogeneouswaveequationthedatamustbeannihilatedbyeachoftheseclosed lightningbolts.Itseemsthatonecouldusethisrestrictionasaguidefornding 64

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aconditioncorrespondingtothepreviousexamplethatguaranteesexistenceand uniquenessforthenon-homogeneousequationposedontherectangle,butweleave thediscussionhereandreturntotheDirichletproblem. 3.2.3ComplexLightningBoltsandtheVekuahull Fromnowon,in C 2 ,weunderstand lightningbolt tomean complexlightningbolt whichwedenebycompleteanalogytotherealcase,butallowingtheverticestohave complexcoordinates.Themethodofannihilatingmeasuressupportedonlightning boltswasrstusedinthecomplexsettingin[46]actuallytheauthorsusedthe name-sets".Noticethat,inthecomplexsetting,Theorem3.2.1anditsproofhold withoutanymodications. WereformulatetheDirichletprobleminastandardwaysimilartowhatwas donewiththewaveequationintheprevioussection.With P x;y apolynomial,let )-385(:= f x;y 2 R 2 : P x;y =0 g bethealgebraiccurvethatcontainsthebounded component,)]TJ/F25 7.9701 Tf 69.754 -1.793 Td [(1 ,wherethedataisposed.Weobtainthe complexication of,denoted ~ ,byallowingcomplexvalues X;Y inthezeroset f X;Y 2 C 2 : P X;Y =0 g Thechangeofvariablesto z = X + iY and w = X )]TJ/F27 11.9552 Tf 12.826 0 Td [(iY convertstheLaplacian to=4 @ 2 @z@w .Solutionsof u =0thentaketheform u z;w = f z + g w wherefandgareholomorphic.TheDirichletproblembecomesataskofmatching thedataon ~ )-308(withasum f z + g w holomorphicina C 2 neighborhoodofasimply connecteddomaincontaining)]TJ/F25 7.9701 Tf 157.922 -1.793 Td [(1 .Forasequenceofpointsin C 2 ,thepropertyofbeing alightningboltdependsheavilyonthecoordinatesystem.Wewillbeinterestedin lightningboltswithrespectto z;w -coordinates. Foradomaininthecomplexplane,the Vekuahull see[63]of,denoted ^ ,isasetin C 2 denedby f z;w : z 2 ;w 2 g ,where := f : 2 g TherelevantpropertyoftheVekuahullforusisthatif f z and g z areanalytic andanti-analyticresp.in,thentheharmonicfunction f z + g w isanalyticin ^ asafunctionoftwocomplexvariables.Thus,failuretoextendasolutionofthe reformulatedDirichletproblemtothe C 2 domain, ^ ,impliesfailuretoextendthe 65

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solutionintherealplaneretrievedbysetting w = z tothedomain. Theorem3.2.2 Let )]TJ/F25 7.9701 Tf 7.314 -1.793 Td [(1 beaconnectedcomponentofthealgebraiccurve )]TJ/F44 11.9552 Tf 7.314 0 Td [(,andlet beasimplyconnecteddomain.Suppose ~ )]TJ/F44 11.9552 Tf 10.845 0 Td [(containsaclosedlightningboltwithrespect tocoordinates z and w oflength 2 n .Supposefurtherthatalong ~ )]TJ/F44 11.9552 Tf 12.044 0 Td [(therearepaths, alsocontainedin ^ ,thatconnecteachvertexto )]TJ/F25 7.9701 Tf 7.314 -1.793 Td [(1 .Then,fortheDirichletproblem on )]TJ/F25 7.9701 Tf 7.314 -1.794 Td [(1 ,thereexistpolynomialdataofdegree n whosesolutioncannotbeanalytically continuedtoallof Proof. Let P z;w bethepolynomialofdegree n furnishedbyTheorem3.2.1. Foreachvertexofthelightningbolt,consideritsconnectingpath"{thepaththat connectsthevertexto)]TJ/F25 7.9701 Tf 123.724 -1.793 Td [(1 .Takeatube-likeneighborhoodofthispaththinenoughto becontainedin ^ .Intersectthiswith ~ )-330(togetastrip", S oftheRiemannsurface, ~ \051,whichcontainsbothanarcof)]TJ/F25 7.9701 Tf 178.546 -1.794 Td [(1 andthevertexunderconsideration.Supposethe solution f z + g z fordata P z; z isanalyticin.Then f z + g w isanalytic in ^ and,inparticular,in S ,sothat P z;w )]TJ/F27 11.9552 Tf 11.72 0 Td [(f z )]TJ/F27 11.9552 Tf 11.72 0 Td [(g w isalsoanalyticin S and, vanishingonanarc)]TJ/F25 7.9701 Tf 117.149 -1.793 Td [(1 ,mustbezeroonallof S .Thus, P z;w = f z + g w at eachvertexofthelightningbolt,implyingthat f z + g w isnotannihilatedbythe alternatingmeasureconstructedinTheorem3.2.1,acontradiction. 3.2.4Ebenfelt'sExampleRevisited LetusapplythemethodofTheorem3.2.2,letting)-285(=)]TJ/F25 7.9701 Tf 293.043 -1.793 Td [(1 betheTVscreen"curve whoseequationis x 4 + y 4 =1.Thesingularitiesdevelopedbythefunctionthatis harmonicinsideandmatchesthedata x 2 + y 2 onthecurve, x 4 + y 4 =1,makeupa discreteinnitesetresidingonthecoordinateaxes.Following[30],onecancalculate thattheclosestsingularitiesaresituatedontheboundaryofthediscofradius2 3 4 In z;w -coordinates, ~ )-303(isgivenbythezerosetof z;w = 1 2 z + w 4 + z )]TJ/F27 11.9552 Tf -424.076 -20.921 Td [(w 4 )]TJ/F19 11.9552 Tf 11.955 0 Td [(8= z 4 +6 z 2 w 2 + w 4 )]TJ/F19 11.9552 Tf 11.955 0 Td [(8,whichcarriestheclosedlightningbolt f ; 1 ; ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1 g : 66

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Usingcoordinates X = z + w 2 and Y = z )]TJ/F28 7.9701 Tf 6.586 0 Td [(w 2 i ,theverticesare v 1 = ; 0, v 2 = ; )]TJ/F27 11.9552 Tf 9.299 0 Td [(i v 3 = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 0,and v 4 = ;i seegure3.6. InordertouseTheorem3.2.2,wemustndpathsalong ~ )-402(connectingeach vertextotherealplane f w = z g = f X 2 R ;Y 2 R g .Noticethat ; 0and )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 0are alreadyintherealplane.In X;Y coordinates,theequationforthecomplexication takesitsoriginalform f X;Y : X 4 + Y 4 =1 g .Considerthecrosssectioncutby R i R X purerealand Y pureimaginary.Writing X = x and Y = iy wesee thatitisacopyoftheslicefromtherealplaneseegure3.6forathreedimensional slicecontainingbothcopies.Connect v 4 = ; )]TJ/F27 11.9552 Tf 9.299 0 Td [(i to v 1 = ; 0using 4 ; 1 t = t;i 4 p 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(t 4 whichtravelsalongtheverticalsliceingure3.6.Wecanalsoconnect v 4 = ; )]TJ/F27 11.9552 Tf 9.298 0 Td [(i to v 3 = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 0usingthepath 4 ; 3 t = )]TJ/F27 11.9552 Tf 9.298 0 Td [(t;i 4 p 1 )]TJ/F27 11.9552 Tf 11.956 0 Td [(t 4 .Similarly,weget paths 2 ; 1 t and 2 ; 3 t intheslice R i R thatconnect v 2 = ; )]TJ/F27 11.9552 Tf 9.299 0 Td [(i to v 1 = ; 0 and v 3 = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 0resp.. Figure3.6:A3-dsliceofthecomplexiedTVscreenthatcontainsallfourverticesand bothconnectingpaths. ThesepathsarecompletelycontainedintheVekuahulloftheopendiscwith anyradiuslargerthan2 3 4 .Indeed,therequirementisthatboth j X + iY j 2 3 4 and j X )]TJ/F27 11.9552 Tf 12.445 0 Td [(iY j 2 3 4 .Considerthepathconnecting v 4 = ; )]TJ/F27 11.9552 Tf 9.298 0 Td [(i to ; 0theother casesaresimilar.Alongthispath, X and iY areeachrealandpositivesothat j X + iY j = X + iY alwaysexceeds j X )]TJ/F27 11.9552 Tf 11.339 0 Td [(iY j .Themaximumof X + iY istakenwhen 67

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X = iY =2 )]TJ/F26 5.9776 Tf 7.782 3.258 Td [(1 4 whichgives X + iY =2 3 4 .Thus,byTheorem3.2.2,thereisquadratic data, P z;w ,forwhichthesolutiontotheDirichletproblemwiththisdatadevelops asingularitynofurtherthan2 3 4 fromtheorigin.Sinceanyquadraticpolynomialcan beobtainedfrom P z;w byadditionofaharmonicquadratic,wecanactuallysay "foranynon-harmonicquadraticdata,thesolutiondevelopsasingularitywithin2 3 4 oftheorigin". Wecanbemorespecicaboutthelocationofthesingularities.Thepaths 4 ; 1 t and 2 ; 1 t areeachcontainedintheVekuahullofanythinneighborhoodof thesegment[ )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 2 3 4 ].Thepaths 4 ; 3 t and 2 ; 3 t arecontainedintheVekuahullof anythinneighborhoodofthesegment[ )]TJ/F19 11.9552 Tf 9.298 0 Td [(2 3 4 ; 1].Therearenosingularitiesinsidethe curve,soweconcludethattherearesingularitiesonthepositiveandnegative x -axis nofurtherfromtheoriginthan2 3 4 .Wecanalsolocatesingularitiesonthe y -axis nofurtherfromtheoriginthan2 3 4 byrepeatingthesestepsforthelightningboltin z;w -coordinates, f i; )]TJ/F27 11.9552 Tf 9.299 0 Td [(i ; )]TJ/F27 11.9552 Tf 9.298 0 Td [(i; )]TJ/F27 11.9552 Tf 9.299 0 Td [(i ; )]TJ/F27 11.9552 Tf 9.298 0 Td [(i;i ; i;i g Itshouldbementionedthat,althoughEbenfelt'smethodismorecomplicated, itproducesanexhaustivedescriptionofthewholeinnitesetofsingularities. Wesummarizeourresultsforthecurve)-434(:= x 4 + y 4 =1,inthefollowing theorem. Theorem3.2.3 FortheDirichletproblemwithanynon-harmonic,quadraticdata onthecurve )-285(:= x 4 + y 4 =1 ,thesolutiondevelopssingularitiesonthepositiveand negativexandyaxesnofurtherfromtheoriginthan 2 3 4 Manyothercurvescontainasimilarclosedlightningbolt.Forinstance,Ebenfeltgeneralizedhisprooftocurveswithcomplexiedform )]TJ/F27 11.9552 Tf 10.442 0 Td [( z 2 k +2 + z k w k + )]TJ/F27 11.9552 Tf 12.313 0 Td [( w 2 k =4,with >> 0.Thesecurvescarryalightningboltwithvertices 1 1 2 k ; 1 1 2 k .WegeneralizeTheorem3.2.3anditsproofinadierentdirection. Theorem3.2.4 Supposepolynomials p x and q x arepositivefor x 2 R + andsatisfy p + q = p + q =1 .ThenfortheDirichletproblemwithnon-harmonic, 68

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quadraticdataonthecurve p x 4 + q y 4 =1 ,thesolutiondevelopssingularitieson thexandyaxesnofurtherfromtheoriginthan max fj x j + j y j : p x 4 + q y 4 =1 g Proof. Thesameclosedlightningbolts f ; 1 ; ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; 1 g and f i; )]TJ/F27 11.9552 Tf 9.298 0 Td [(i ; )]TJ/F27 11.9552 Tf 9.299 0 Td [(i; )]TJ/F27 11.9552 Tf 9.299 0 Td [(i ; )]TJ/F27 11.9552 Tf 9.299 0 Td [(i;i ; i;i g inz,w-coordinateslieonthecomplexicationof p x 4 + q y 4 =1.In X;Y -coordinatestheseare f ; 0 ; ; )]TJ/F27 11.9552 Tf 9.299 0 Td [(i ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; 0 ; ;i g and f ; 1 ; )]TJ/F27 11.9552 Tf 9.299 0 Td [(i; 0 ; ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; i; 0 g .Thecurve, p x 4 + q y 4 =1,alsohasthesamesymmetryas theTVscreeninthe X purerealand Y purerealdslices.Thus,wecanconstruct connectingpathssimilarto i;j t bytravelingalongthe R i R sliceorthe i R R sliceforthesecondlightningbolt.Alongthepathsin R i R X + iY and X )]TJ/F27 11.9552 Tf 12.215 0 Td [(iY arepurereal,andboundedby max fj x j + j y j : p x 4 + q y 4 =1 g .Alongthepathsin the i R R slice,thesameestimateholdsbut X + iY and X )]TJ/F27 11.9552 Tf 11.067 0 Td [(iY arepureimaginary. Weusethinneighborhoodsofintervalsonthe x and y axestoobtainonechoiceof foreachlightningbolt.TheproofisnishedbyanapplicationofTheorem3.2.2. Remark: ItisnaturaltochoosediscsforinThereom3.2.2ifweareonlyinterestedinboundingthedistanceofasolution'ssingularitiestotheinitialcurve.If wewantmoredetailedinformation,wecanuseadditionallightningbolts,alternative connectingpaths,andchoicesforwhoseVekuahullsmorefrugallycatchtheconnectingpaths.InprovingTheorems3.2.3and3.2.4,morefrugally"meantchoosing tobeathinneighborhoodofasegmentcontainingthe z and w projectionsofthe connectingpath.Generally,though,the z and w projectionsofaconnectingpath formtwodierentpathswithcommonendpoints,sothatchoosingtobeathin neighborhoodviolatestheassumptioninTheorem3.2.2thatissimplyconnected. 3.2.5AFamilyofCurvesNotCoveredbyRender'sTheorem Weconsiderthezerosetsofthefollowingfamilyofcubicperturbationsoftheunit circlewhichhaveanovalcomponentandanunboundedcomponentseegure3.7. P x;y = x 2 + y 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 )]TJ/F19 11.9552 Tf 11.956 0 Td [(2 x 3 + xy 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(x 2 + y 2 .2.12 69

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Figure3.7:Aplotfortheperturbedcirclewithunboundedcomponent: = : 1 Thisfamilyischosenfollowingtheideain[14]sothatthecomplexication ofthisfamilyofcurvestakesthesimpleform z;w = "z 2 )]TJ/F27 11.9552 Tf 11.985 0 Td [(z "w 2 )]TJ/F27 11.9552 Tf 11.984 0 Td [(w )]TJ/F19 11.9552 Tf 11.985 0 Td [( "z 2 )]TJ/F19 11.9552 Tf -424.076 -20.922 Td [(1 "w 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1. Rewrite z;w =0as "z 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(z "z 2 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 "w 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(w "w 2 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 =1andabbreviateas z w =1, where z = "z 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(z "z 2 )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 .Choosinganycomplexnumber a andsolving z = a and w = 1 a givescoordinates z a ;w 1 a forapointonthecomplexiedcurve.Since wegenerallygettwosolutionseach,thereareinnitelymanyclosedlightningbolts. Forinstance,with a = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1= 1 a z = w = 1 p 1+8 4 ,givingtheclosedlightningbolt abbreviate1+8 with 1 )]TJ 11.955 8.694 Td [(p 4 ; 1 )]TJ 11.955 8.694 Td [(p 4 ; 1 )]TJ 11.955 8.694 Td [(p 4 ; 1+ p 4 ; 1+ p 4 ; 1+ p 4 ; 1+ p 4 ; 1 )]TJ 11.955 8.694 Td [(p 4 Therstvertexisintherealplane f w = z g ontheboundedcomponentof z; z =0.Thelastis,incidentally,intherealplaneontheunboundedcomponent. Thisconformstoourintuitionthatthepresenceofasecondcomponentofthezero setisanobstacletoanalyticcontinuationofsolutions.Forinstance,ifasolution totheDirichletproblemposedononecomponentofanirreduciblecurveisentire 70

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thenitmustmatchthedataonanyothercomponents,accidentally"solvingan overdeterminedboundaryvalueproblem. Weproducepathsconnectingtheverticestotheboundedcomponentofthe curvebyrstparameterizingasinglepath a t andthenconsidering z a t ;w 1 a t and z 1 a t ;w a t .Ourpath a t willbeaclosedpathwith a = a = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1. As a t traversesthispath,theexpression1 )]TJ/F19 11.9552 Tf 12.966 0 Td [(4 "a )]TJ/F27 11.9552 Tf 12.966 0 Td [(a insidethesquareroot intheformulafor z a = w a willwindexactlyoncearoundzero,switchingthe branchofsquareroot.Theresultingpathsdependontheinitialchoiceofthebranch ofthesquareroot.Weusethepaths P ; t = z a t ;w 1 a t and Q ; t = z 1 a t ;w a t ,wherethesubscriptsindicatewhichbranchofthesquarerootis usedat t =0toobtain z )]TJ/F19 11.9552 Tf 9.299 0 Td [(1and w )]TJ/F19 11.9552 Tf 9.299 0 Td [(1+"indicatesthebranchwhichispositive forpositivereals.Thesequenceofpaths Q )]TJ/F28 7.9701 Tf 6.586 0 Td [(; )]TJ/F19 11.9552 Tf 7.084 1.793 Td [( t ;P )]TJ/F28 7.9701 Tf 6.586 0 Td [(; + t ;Q + ; + t ;P + ; )]TJ/F19 11.9552 Tf 7.085 1.793 Td [( t connect theverticesconsecutively,andultimatelyeachvertextotherst,whichresideson theboundedcomponentofthecurveintherealplane, f w = z g Noticethat1 )]TJ/F19 11.9552 Tf 10.528 0 Td [(4 "a )]TJ/F27 11.9552 Tf 10.528 0 Td [(a hastwozeros 1 2 i q 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1.Have a t travelalong theunitcirclefrom )]TJ/F19 11.9552 Tf 9.299 0 Td [(1to )]TJ/F27 11.9552 Tf 9.298 0 Td [(i .Thenfromtherealongtheimaginaryaxisto )]TJ/F28 7.9701 Tf 6.587 0 Td [(i 2 q 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 andnallyto 1 2 )]TJ/F27 11.9552 Tf 12.247 0 Td [(i q 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1wherewewillreplacethisendpointwithatinycircle arounditbeforeretracingourstepsbackto a = )]TJ/F19 11.9552 Tf 9.298 0 Td [(1.Since a t windsaroundaroot ofmultiplicityone,theimage,1 )]TJ/F19 11.9552 Tf 12.391 0 Td [(4 "a t )]TJ/F27 11.9552 Tf 12.391 0 Td [(a t windsaroundtheoriginexactly oncebytheargumentprinciple,switchingbranchesofthesquareroot.As a traces thisarc, 1 a travelsalongtheunitcirclefrom )]TJ/F19 11.9552 Tf 9.298 0 Td [(1to i thenalongtheimaginaryaxis to2 i p 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(" andfromthereto 1 2 )]TJ/F27 11.9552 Tf 12.041 0 Td [(i q 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 .Weneedtoestimate j )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 a j and j )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 1 a j alongthispath. Forthepiecealongtheunitcircle,weusethefactthat a hasmodulus1and distancefrom1atleast p 2.Thus, j z a j = j 1 p 1 )]TJ/F25 7.9701 Tf 6.586 0 Td [(4 "a )]TJ/F28 7.9701 Tf 6.587 0 Td [(a 2 )]TJ/F28 7.9701 Tf 6.587 0 Td [(a j 1+ p j 1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(4 "a )]TJ/F28 7.9701 Tf 6.586 0 Td [(a j 2 p 2 1+ p 1+8 2 p 2 .Wegetthesameestimatefor j )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 1 a j since,alongthispartofthepath, 1 a alsohasmodulus1anddistancefrom1atleast p 2. Alongthepathwhere a and 1 a arepureimaginary,theyeachhavedistancefrom 1atleast1.Sincewealsohave j 1 a j 1,writing 1 a = xi with x 2 ; 1],weestimate 71

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j )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 1 a j 1+ p j 1 )]TJ/F25 7.9701 Tf 6.586 0 Td [(4 "xi )]TJ/F25 7.9701 Tf 6.586 0 Td [(4 "x 2 j 2 1+ p 1+4 2 .Similarly,since a 2 [ )]TJ/F28 7.9701 Tf 6.586 0 Td [(i 2 q 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ; )]TJ/F27 11.9552 Tf 9.299 0 Td [(i ]write a = )]TJ/F27 11.9552 Tf 9.298 0 Td [(ix with x 2 [1 ; 1 2 q 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1].Then j )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 a j 1+ p j 1+4 "xi )]TJ/F25 7.9701 Tf 6.587 0 Td [(4 "x 2 j 2 1+ p j 1+4 "xi j 2 1+ p 1+2 p 2 Alongthelastbitofsegment, a = 1 2 x )]TJ/F27 11.9552 Tf 12.829 0 Td [(i q 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1,with x 2 [0 ; 1].Since forsmall j a )]TJ/F27 11.9552 Tf 12.079 0 Td [(a j = j x 2 )]TJ/F28 7.9701 Tf 13.275 4.707 Td [(x 2 + 1 4 + x )]TJ/F19 11.9552 Tf 12.079 0 Td [(1 i 2 q 1 )]TJ/F19 11.9552 Tf 11.956 0 Td [(1 j isgreatestwhen x =0, wecanusethesameestimatefor j )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 a j aswedidalongtheimaginaryaxis.We canalsousethesameestimateforthenumeratorof j )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 1 a j .Forthedenominator, 2 j 1 )]TJ/F25 7.9701 Tf 31.299 4.707 Td [(2 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [(i p 1 )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 j =2 j 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 + i q 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 j 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 Thus,theconnectingpathsarecontainedintheVekuahullofthedisccentered attheoriginwithradius 1+ p 1+2 p 2 )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 .Wehavenowproventhefollowing. Theorem3.2.5 FortheDirichletproblemwithanynon-harmonic,quadraticdata posedontheboundedcomponentof x 2 + y 2 )]TJ/F19 11.9552 Tf 11.187 0 Td [(1 )]TJ/F19 11.9552 Tf 11.187 0 Td [(2 x 3 + xy 2 )]TJ/F27 11.9552 Tf 11.188 0 Td [(x 2 + y 2 =0 small, thesolutiondevelopssingularitiesnofurtherfromtheoriginthan 1+ p 1+2 p 2 )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 ,which is,asymptotically,twicethedistancefromtheboundedcomponenttotheunbounded component. Relevanttoourstudy,authorsin[22]investigatedwhenpolynomialsolutions areguaranteedbypolynomialdata.Theygivethenecessarybutbynomeanssufcientconditionthattheseniortermofthecurve'sequationmustdivideahomogeneousharmonicpolynomial.Thecubic,8 x x 2 )]TJ/F27 11.9552 Tf 12.645 0 Td [(y 2 +57 x 2 +77 y 2 =49see gure3.8satisesthisnecessarycondition,since8 xy x 2 )]TJ/F27 11.9552 Tf 13.047 0 Td [(y 2 isahomogeneous harmonic.Moreover,8 xy x 2 )]TJ/F27 11.9552 Tf 12.739 0 Td [(y 2 solvestheDirichletproblemforthecubicdata g x;y = y [57 x 2 +77 y 2 )]TJ/F19 11.9552 Tf 11.956 0 Td [(49]. In z;w -coordinates,thecomplexiedformis 2 z + w z 2 + w 2 +67 zw )]TJ/F19 11.9552 Tf 11.955 0 Td [(5 z 2 + w 2 =49 ; whichhasanontrivialsliceintheplanewith z and w purereal.Thecurveinthis planeincludesanovalcomponentseegure3.9containingallsixverticesofthe 72

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Figure3.8:Amapleplotofthecubic8 x x 2 )]TJ/F49 10.9091 Tf 10.264 0 Td [(y 2 +57 x 2 +77 y 2 =49,showingthebounded componentandoneunboundedcomponenttherearetwootherunboundedcomponents furtheraway. closedlightningbolt, f )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(7 = 2 ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(7 ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(7 = 2 ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(7 ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(7 ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(7 = 2 ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(7 ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(7 = 2 ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 g : Weimmediatelyobtainpathsconnectingtheverticestotheboundedcomponentof thereal f w = z g curvebytravelingalongtheovaltothepoint )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(1whichis ontheboundedcomponentoftherealcurve.The z and w projectionsoftheoval areeachcontainedintheinterval[ )]TJ/F19 11.9552 Tf 9.298 0 Td [(7 : 622 ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(1].Wecanchooseathinneighborhood ofthe x -axissegment[ )]TJ/F19 11.9552 Tf 9.299 0 Td [(7 : 622 ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(1]forourdomaininTheorem3.2.2.Thus,travelingalongthenegative x -axisweencounterasingularity,ifnotbeforecrossingthe unboundedcomponentat x =-7,nofurtherthan10%ofthedistancefromthe boundedcomponenttotheunboundedcomponent.Wesummarizethisnalresult. Theorem3.2.6 FortheDirichletproblemposedinsidetheboundedcomponentofthe perturbedellipse, 8 x x 2 )]TJ/F27 11.9552 Tf 11.538 0 Td [(y 2 +57 x 2 +77 y 2 =49 ,thereexistcubicdataforwhichthe solutiondevelopsasingularityonthex-axisnofurtherfromtheoriginthan 7 : 622 comparetothex-intercept, )]TJ/F19 11.9552 Tf 9.299 0 Td [(7 ; 0 ,ofthenearestunboundedcomponent. Remarks: Onehopesforatheoremgivingtheexistenceofclosedlightningboltson thecomplexicationofallirreduciblealgebraiccurvesofdegreegreaterthantwothat 73

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Figure3.9:Alightningboltwithsixverticesonthecubic2 z + w z 2 + w 2 +67 zw )]TJ/F15 10.9091 Tf 9.881 0 Td [(5 z 2 + w 2 =49intheplanewith z and w real. 74

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haveaboundedcomponent.Thiswouldproveintwodimensionstheconjectureof KhavinsonandShapirothattheellipseistheonlycurveforwhichentiredatagives entiresolutions.Generaltheoremsontheexistenceofclosedlightningboltsaregiven in[14],[15],buttheyrelyheavilyonthehypothesizedformoftheRiemannmapofthe interiordomain.Thus,amodiedconstructionisneededformoregeneralalgebraic curves. 3.3TheKhavinson-ShapiroConjectureandPolynomialDecompositions Thissectionistakenfromthepaper[79]writtenjointlywithHermannRenderwhich waspublishedinTheJournalofMathematicalAnalysisanditsApplications. Themainresultstatesthefollowing:Let beapolynomialin n variables. Supposethatthereexistsaconstant C> 0suchthatanypolynomial f hasapolynomialdecomposition f = q f + h f with k h f =0anddeg q f deg f + C: Then deg 2 k .Here k isthe k thiterateoftheLaplaceoperator : Asanapplication,newclassesofdomainsin R n areidentiedforwhichtheKhavinson-Shapiro conjectureholds. Itisinstructivetoconsiderthecase k =1inthestatementabove. 3.3.1AlgebraicDirichletproblemsandPolyharmonicDecompositions Areal-valuedfunction h denedonanopenset U in R n iscalled k harmonic or polyharmonicoforder k if h isdierentiableuptotheorder2 k andsatisesthe equation k h x =0forall x 2 U: HeredenotestheLaplacian @ 2 @x 2 1 + :::: + @ 2 @x 2 n and k isthe k thiterateoftheLaplaceoperator : Polyharmonicfunctionshavebeenstudied extensivelyin[8],andtheyareusefulinmanybranchesinmathematics,see[68]. Forexample,inelasticitytheoryanddynamicsofslow,viscousuidspolyharmonic functionsoforder2 ; ormorebriey, biharmonicfunctions ,areveryimportant. Beforediscussingourmainresultswestillneedsomenotation.By R [ x 1 ;:::;x n ] wedenotethespaceofallpolynomialswithrealcoecientsinthevariables x 1 ;:::;x n : 75

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Frequentlyweusethefactthatanypolynomial ofdegree m canbeexpandedintoa sumofhomogeneouspolynomials j ofdegree j for j =0 ;:::;m ,andwewriteshortly = 0 + ::: + m ;here m 6 =0iscalledthe principalpart or leadingpart ofthe polynomial : Thedegreeofapolynomial isdenotedbydeg : Inthisarticlewewillbeconcernedwithaconjectureseebelowwhicharises naturallyfromofthefollowingstatementprovenin[86,Theorem3]for k =1see also[10]: Theorem3.3.1 Let 2 R [ x 1 ;:::;x n ] beapolynomialofdegree 2 k suchthatthe leadingpart 2 k isnon-negative.Thenforanypolynomial f 2 R [ x 1 ;:::;x n ] thereexist uniquepolynomials q f and h f in 2 R [ x 1 ;:::;x n ] suchthat f = q f + h f and k h f =0 : .3.13 Moreover,thedecompositionisdegreepreserving,meaningthat deg h f deg f and, consequently, deg q f deg f )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 k: Theorem3.3.1isrelatedtothepolynomialsolvabilityofDirichlet-typeproblems.Forexample,letusconsiderthepolynomial 0 x = n X j =1 x 2 j a 2 j )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 ; .3.14 so E 0 := x 2 R d : 0 x < 0 isanellipsoid.Thenthedecomposition.3.13 where k =1showsthewellknownandoldfactthatforanypolynomial f ,restrictedtotheboundary @E 0 ; thereexistsaharmonic polynomial h whichcoincides withthedatafunction f on @E 0 .Inotherwords,thesolutionsforpolynomialdata functionsoftheDirichletproblemfortheellipsoidareagainpolynomials,see[9],[12], [22],or[64]. In[64]D.KhavinsonandH.S.Shapiroformulatedthefollowingtwoconjectures iandiiforboundeddomainsforwhichtheDirichletproblemissolvable: 76

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KS:isanellipsoidifforeverypolynomial f thesolutionoftheDirichlet problem u f isiapolynomialand,respectively,iientire. Conjecturesiandiiarestillopen,butimportantcontributionshavebeen madebyseveralauthors.Mostoftheresultsareprovenforthetwo-dimensional case,seee.g.[22],[30],[77]and[46].M.PutinarandN.Stylianopouloshaveshown recentlyin[85]thattheconjectureiforasimplyconnectedboundeddomainin thecomplexplaneistrueifandonlyiftheBergmanorthogonalpolynomialssatisfy aniterecurrencerelation.D.KhavinsonandN.Stylianopoulosprovedamongother thingsthattheBergmanorthogonalpolynomialssatisfyarecurrencerelationoforder N +1ifandonlyifconjectureiholdsandadegreeconditionforthesolution u f issatised,fordetailsandfurtherdiscussionsee[65].In[86]H.Renderhasgiven asolutionforiandiiforarbitrarydimensionandforalargebutnotexhaustive classofdomains. Webelievethatthevalidationofthefollowingconjectureforthecase k =1 wouldbeanimportantstepforprovingtheKhavinson-Shapiroconjecturee.g.confer theproofofTheorem27in[86]: Conjecture3.3.2 Suppose 2 R [ x 1 ;:::;x n ] isapolynomial,suchthateverypolynomial f 2 R [ x 1 ;:::;x n ] hasadecomposition f = q f + h f ,where h f ispolyharmonicof order k .Then deg 2 k Weareabletoprovetheconjectureifweaddadegreeconditionontheinvolved polynomialswhichisinthespiritoftheabove-mentionedwork[65].Moreprecisely, themainresultofthepresentpaperisthefollowing: Theorem3.3.3 Let 2 R [ x 1 ;:::;x n ] beapolynomial.Supposethatthereexistsa constant C> 0 suchthatforanypolynomial f 2 R [ x 1 ;:::;x n ] thereexistsadecomposition f = q f + h f with k h f =0 and deg q f deg f + C: .3.15 Then deg 2 k 77

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Theorem3.3.3willbeaconsequenceofasomewhatstrongerresultprovedafter ashortdiscussionofharmonicdivisors.Inpassingwenotethattheconjecture3.3.2 doesnotholdforpolynomials withcomplexcoecients,see[55]. Itisanaturalquestiontoaskunderwhichconditionsatthegivenpolynomial x thedegreeconditioninTheorem3.3.3isautomaticallysatised.Inotherwords, canweconcludefromtheequation f = q f + h f with k h f =0 thatarestrictionmustexistonthedegreeof q f or h f intermsofthedegreeof f ? Forthecase k =1weshallproveinSection4thatthedegreecondition.3.15is satisedifitheleadingpart t of containsanon-negativenon-constantfactor orii hasahomogeneousexpansionoftheform = t + s + ::: + 0 where s 6 =0containsanon-negative,non-constantfactor.Anextensionoftheseresultsfor arbitrary k isalsogiven.Theseresultsallowustoidentifynewtypesofdomainsin R n forwhichtheKhavinson-Shapiroconjectureistrue. 3.3.2Fischeroperatorsandharmonicdivisors For Q 2 R [ x 1 ;:::;x n ]letusdene Q D asthedierentialoperatorreplacingamonomial x appearingin Q bythedierentialoperator @ =@x ,where isamulti-index. Fortwopolynomials Q and wecalltheoperator F Q : R [ x 1 ;:::;x n ] R [ x 1 ;:::;x n ] denedby F Q q := Q D q q 2 R [ x 1 ;:::;x n ].3.16 the Fischeroperator ;forthesignicanceofthisnotionwerefertotheexcellent exposition[95],or[10],[86].WeshallneedthefollowingresultduetoE.Fischer[39] whichisinaslightlymodiedformvalidforpolynomialswithcomplexcoecients, see[95]: Theorem3.3.4 Let Q 2 R [ x 1 ;:::;x n ] beahomogeneouspolynomial.Thentheoperator q 7)167(! Q D Qq isbijective. 78

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Atrstweobservethattheconjecture3.3.2isequivalenttothesurjectivity oftheFischeroperatorwith Q = P n i =1 x 2 i k ;thisfactiswellknown,butforthe convenienceofthereader,weincludetheshortproof. Proposition3.3.5 Suppose k 2 N and isapolynomial.Theoperator F k q := k q issurjectiveifandonlyifeverypolynomial f canbedecomposedas f = q f + h f where h ispolyharmonicoforder k: Proof. Taking k ofbothsidesof f = q + h gives k f = F k q .Given g wecannd f suchthat g = k f ,showing F k issurjective.Conversely,if F k issurjective,then given f thereisa q suchthat k f = F k q ,showingthat h = f )]TJ/F27 11.9552 Tf 11.015 0 Td [(q ispolyharmonic oforder k: Apolynomial f m iscalled homogeneous ofdegree m if f m rx = r m f m x for all r> 0andforall x 2 R n .Wewilluse P N todenotethespaceofpolynomials ofdegreeatmost N ,and P N hom thespaceofhomogeneouspolynomialsofdegree N Forahomogeneouspolynomial wedenethespaceofallhomogeneous k -harmonic divisorsofdegree m of by D m k = q 2 P m hom : k q =0 : For k =1weobtainthedenitionofaharmonicdivisorofdegree m whicharises intheinvestigationofstationarysetsforthewaveandheatequation,see[4],[5],and theinjectivityofthesphericalRadontransform,see[6],[3]. Itisaninterestingbutdicultproblemtocomputethedimensionofthespace D m k andtodescribehowitdependsonthepolynomial : Intheproofofourmain resultTheorem3.3.3weshallusetheroughupperestimateprovidedinthenext propositionandtheremarksfollowing: 79

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Proposition3.3.6 Let 2 R [ x 1 ;:::;x n ] beahomogeneouspolynomial.Then dim D m k dim f 2 P m hom : k f =0 : Proof. Let q 2 D m k : Then q 2 P m hom and q = h forsome h 2 P m + t hom with k h =0, where t isthedegreeof .Clearlywehave D q = D h and 0= D )]TJ/F19 11.9552 Tf 5.479 -9.684 Td [( k h = k D h : .3.17 ByTheorem3.3.4,theoperator F denedby F q = D q isbijective,and from q = h weinferthat q = F )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 w with w := D h: Equation.3.17shows that w 2 f 2 P m hom : k f =0 : Thus D m k F )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 \010 f 2 P m hom : k f =0 : Since F )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 isabijectiveoperator,theclaimisnowobvious. Letusdene H m k := f 2 P m hom : k f =0 : ByTheorem3.3.4for Q x = j x j 2 k itfollowsthatanypolynomial f hasaFischerdecomposition f = j x j 2 k q + h where h is k -harmonic.Moreover, h and q arehomogeneousi f is.Sowehave P m hom = j x j 2 k P m )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 k hom H m k : Thusweobtain dim D m k dim H m k =dim P m hom )]TJ/F19 11.9552 Tf 11.291 0 Td [(dim P m )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 k hom : .3.18 ThefollowingquestionwasposedbyM.Agranovskyforthecase k =1in[3], whereitwasalsoansweredinthecasethat factorscompletelyintolinearfactors. Question3.3.7 Whatistheasymptoticbehaviorof dim D m k ,as m !1 ? 80

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Weexpectthatafullanswertothisquestionwouldallowustorelaxtheassumption ondegreeappearinginTheorem3.3.3. 3.3.3Proofofthemainresult Assumethat2 k t andlet beapolynomialofdegree t andlet F k betheFischer operatordenedinProposition3.3.5.Thefollowingtechnicalnotionisacrucialtool forprovingourmainresultTheorem3.3.3:Foranaturalnumber M dene S i P i asthesubspacewhoseimageunder F k iscontainedin P M + t )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 k ,i.e., S i := f q 2 P i : F k q 2 P M + t )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 k g for i 2 N 0 : Since hasdegree t itfollowsthat P M = S M S M +1 ::: S M + j forall j 1 : Proposition3.3.8 Let = t + ::: + 0 beapolynomialofdegree t andlet M be anaturalnumber.Thenforall j 2 N dim S M + j dim S M + j )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 +dim D M + j k t : Proof. Forgiven j 2 N wewillconstructaspace Q j suchthat S M + j = S M + j )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 Q j anddim Q j dim D M + j k t .Firstdene Q H;j := f q M + j : q M + j isthedegreeM + j homogeneoustermofsome q 2 S M + j g .Choosenitelymanypolynomialsin S M + j whoseleadingtermsformabasisfor Q H;j ,anddene Q j tobethesubspaceof S M + j spannedbythesepolynomials.Suppose^ q 2 S M + j .ThedegreeM + j homogeneous term^ q M + j possiblyzerocanbematchedbytheleadinghomogeneoustermofsome q 2 Q j sothat^ q )]TJ/F27 11.9552 Tf 11.955 0 Td [(q 2 S M + j )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 .Thisshowsthat S M + j = S M + j )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 Q j Now,wewillestablishdim Q j dim D M + j k t .Itsucestoshowthat Q H;j D M + j k t ,sincedim Q j =dim Q H;j byconstruction.Suppose q M + j 2 Q H;j 81

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isnonzero,i.e.,thereisa q 2 S M + j anddeg q = M + j suchthat q M + j istheleading homogeneoustermof q .Since F k q 2 P M + t )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 k ,wehavedeg k q M + t )]TJ/F19 11.9552 Tf 11.303 0 Td [(2 k Thisimpliesthattheleadingterm, k t q M + j ,of k q iszerosinceithasdegree M + j + t )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 k .i.e., t q M + j is k -harmonic.Therefore, Q H;j D M + j k Themainresultofthispaper,Theorem3.3.3,followsnowfromthefollowing moregeneralresultbytaking =1: Theorem3.3.9 Let beapolynomial.Supposethatthereexistconstants 1 C> 0 suchthatforanypolynomial f thereexistsadecomposition f = q f + h f with k h f =0 and deg q f deg f + C: Then deg 2 k n )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 : Proof. Let t bethedegreeof ,andsuppose t 2 k .If t< 2 k ,thereisnothingto prove.Let f 2 P M + t )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 k andsupposethat M> 2 k: Chooseapolynomial g 2 P M + t with k g = f: Byassumptionthereexists q f and h f with g = q f + h f and k h f =0 anddeg q f M + t + C: Then f = k g = F k q f andweinfertheinclusion P M + t )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 k F k )]TJ/F41 11.9552 Tf 5.48 -9.684 Td [(P B M .3.19 with B M := M + t + C M: Usingtheabovenotation S B M = f q 2 P B M : F k q 2 P M + t )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 k g weseethat.3.19impliesthat P M + t )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 k F k S B M : Since F k isalinearoperator,wehave dim P M + t )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 k dim F k S B M dim S B M : .3.20 ApplyingProposition3.3.8inductivelyweobtain dim S B M dim P M + B M X j = M +1 dim D j k t .3.21 82

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Since P M + t )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 k = P M P M +1 hom ::: P M + t )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 k hom anddim P M +1 hom dim P M + j hom for j 1we inferfrom.3.20and.3.21theinterestingformula t )]TJ/F19 11.9552 Tf 11.956 0 Td [(2 k dim P M +1 hom B M X j = M +1 dim D j k t : .3.22 Furtherweknowfrom.3.18thatdim D j k t dim P j hom )]TJ/F19 11.9552 Tf 12.545 0 Td [(dim P j )]TJ/F25 7.9701 Tf 6.586 0 Td [(2 k hom : Thusthe righthandsidein.3.22isatelescopingsum.Usingthatdim P j hom dim P B M hom for j = B M )]TJ/F19 11.9552 Tf 12.206 0 Td [(2 k +1 ;::::;B M anddim P M +1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 k hom dim P j hom forthelowerindiceswecan estimate B M X j = M +1 dim D j k t 2 k dim P B M hom )]TJ/F19 11.9552 Tf 11.956 0 Td [(2 k dim P M +1 )]TJ/F25 7.9701 Tf 6.587 0 Td [(2 k hom : Thusweinferfrom.3.22andthewellknownfact dim P M +1 hom = n + M n )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 = n + M M +1 ; provenin[9]that t )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 k M +2 ::: M + n n )]TJ/F19 11.9552 Tf 11.955 0 Td [(1! 2 k B M +1 ::: B M + n )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 )]TJ/F19 11.9552 Tf 11.955 0 Td [( M +2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 k :::: M + n )]TJ/F19 11.9552 Tf 11.956 0 Td [(2 k n )]TJ/F19 11.9552 Tf 11.955 0 Td [(1! Clearlytheterm n )]TJ/F19 11.9552 Tf 11.955 0 Td [(1!canbecanceledintheinequality.Dividetheinequalityby M n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 onbothsidesandrecallthat B M = M + t + C .Nowtakethelimit M !1 andweobtain t )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 k 2 k )]TJ/F27 11.9552 Tf 5.48 -9.684 Td [( n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 : Thisimplies t 2 k n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 andtheproofiscomplete. 3.3.4Criteriafordegree-relateddecompositions Wearenowturningtothequestionunderwhichconditionsthedegreeconditionis automaticallysatised.Therstcriterionissimpletoprove: 83

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Proposition3.3.10 Supposethat isapolynomialofdegree t> 2 and = t + ::: + 0 isthedecompositionintoasumofhomogeneouspolynomials.Assumethe polynomial t containsanon-negative,non-constantfactor.Let f beapolynomial andassumethatthereexistsadecomposition f = q + h where h isharmonicand q isapolynomial.Then deg q deg f )]TJ/F27 11.9552 Tf 10.52 0 Td [(t and deg h deg f: Proof. Write q = q M + ::: + q 0 withhomogeneouspolynomials q j ofdegree j =0 ;:::;M: Expandtheproduct q intoasumofhomogeneouspolynomials,so q = t q M + R x where R x isapolynomialofdegree deg f: Since f = q weconcludethat t q M =0 ; so t q M isharmonic.BytheBrelotChoquettheorem,aharmonicpolynomialcannothavenon-negativefactors,see[20]. Thus t q M =0,andweobtainacontradiction. Thenextcriterionismorediculttoproveandusesagainideasfromtheproof oftheBrelot-Choquettheorem: Theorem3.3.11 Supposethat isapolynomialofdegree t> 2 and = t + s + s )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 + ::: + 0 isthedecompositionintoasumofhomogeneouspolynomials.Assume thepolynomial s isnon-zeroandcontainsanon-negative,non-constantfactor.Let f beapolynomialandassumethatthereexistsadecomposition f = q + h where h isharmonicand q isapolynomial.Then deg q 2 )]TJ/F27 11.9552 Tf 11.986 0 Td [(s +deg f and deg h t +2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(s +deg f: BeforeprovingTheorem3.3.11wenoticethefollowingconclusion: 84

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Corollary3.3.12 Supposethat isapolynomialwithanon-zerosecond-highest degreetermthatcontainsanon-negativefactor.Ifeverypolynomial f hasaFischer decomposition f = q f + h f with h f harmonic,then deg 2 Proof. Supposedeg > 2.ByTheorem3.3.11,deg q f )]TJ/F19 11.9552 Tf 11.598 0 Td [(deg f isbounded.Nowwe canapplyTheorem3.3.3,toobtaindeg 2. ThefollowinglemmaisneededfortheproofofTheorem3.3.11: Lemma3.3.13 Supposethat isapolynomialofdegree t> 2 and = t + s + s )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 + ::: + 0 isthedecompositionintoasumofhomogeneouspolynomials.Assume that g 2 P m and q isapolynomialofdegree M suchthat F k q := q = g and M + s>m .Thenforevery p 2 P s )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 Z S n )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 q 2 M s pd =0 ; where q M 6 =0 istheleadinghomogeneouspartof q Proof. ofLemma:Write q = q M + ::: + q 0 withhomogeneouspolynomials q j ofdegree j =0 ;:::;M: Expandtheproduct q intoasumofhomogeneouspolynomials, q = t q M + ::: + t q M )]TJ/F28 7.9701 Tf 6.587 0 Td [(t + s +1 + t q M )]TJ/F28 7.9701 Tf 6.587 0 Td [(t + s + s q M + R x .3.23 where R x isapolynomialofdegree m ,we concludethat t q M =0and t q M )]TJ/F28 7.9701 Tf 6.587 0 Td [(t + s + s q M =0.Thus,wecanwrite t q M = h M + t .3.24 t q M )]TJ/F28 7.9701 Tf 6.587 0 Td [(t + s + s q M = h M + s ; .3.25 where h M + t and h M + s arehomogeneousharmonicpolynomials. 85

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Take p 2 P s )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ,andmultiplyequation.3.25by q M p andintegrateoverthe unitsphere, S n )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 .Then Z S n )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 t q M )]TJ/F28 7.9701 Tf 6.586 0 Td [(t + s q M pd + Z S n )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 s q 2 M pd = Z S n )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 h M + s q M pd: Sincedeg q M p deg f +2 : Wehave q = f and M + s>deg f .Then, q satisfyLemma3.3.13with g = f ,andthus R S n )]TJ/F26 5.9776 Tf 5.756 0 Td [(1 q 2 M s pd =0 ; forall p ofdegree 2 ; inparticular doesnotcontainanonnegative non-constantfactor,see[20].Weperturbtheequationfortheunitball j x j 2 )]TJ/F19 11.9552 Tf 12.368 0 Td [(1by "' ,i.e.weconsider x := j x j 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(1+ "' x for "> 0 : .3.26 86

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If "> 0issmallenough,thenthecomponentof E := f < 0 g containing0is aboundeddomainin R d : ThentheDirichletproblemforthedatafunction j x j 2 = x 2 1 + ::: + x 2 n restrictedto @E hasaharmonicpolynomialsolution u f x =1 )]TJ/F27 11.9552 Tf 11.308 0 Td [("' x since j x j 2 = x 1+1 )]TJ/F27 11.9552 Tf 11.955 0 Td [("' x : Notethatinthisexamplethedegreeofthesolution u f fortheDirichletproblemis higherthanthedegreeofthedatafunction f: Thequestionariseswhetheranypolynomialdatafunctionmayhaveapolynomialsolution.Ifthisisthecase,and isirreducibleandchangesthesignin aneighborhoodofsomepointin @E thentheproofofTheorem27in[86]implies thatforanypolynomial f thereexistsadecomposition f = q f + h f where h f is harmonic.ByCorollary3.3.12deg 2 : Thuswehaveprovedthatforthisclassof examplestheKhavinson-Shapiroconjectureistrue. IntherestofthissectionweextendTheorem3.3.11tothecase k 1 : We considerthefollowinginnerproduct h f;g i := Z R n f x g x e j x j 2 dx .3.27 andthefollowingorthogonalityconditionestablishedin[86]. Theorem3.3.14 Supposethat f isahomogeneouspolynomial,andlet k 2 N with 2 k )]TJ/F19 11.9552 Tf 12.03 0 Td [(1 deg f .Then k f =0 ifandonlyif h f;g i =0 forallpolynomials g with 2 k )]TJ/F19 11.9552 Tf 11.956 0 Td [(1+ deg g< deg f Theorem3.3.15 Supposethat isapolynomialofdegree t>s and = t + s + s )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 + ::: + 0 isthedecompositionintoasumofhomogeneouspolynomials.Assume thepolynomial s 6 =0 isnon-negative.Ifthepolynomial f hasthedecomposition f = q + h where h is k -harmonic,then deg q 2 k )]TJ/F27 11.9552 Tf 11.955 0 Td [(s +deg f 87

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Proof. Supposethat M + s> 2 k +deg f ,where f = q + h and M =deg q Wewillderiveacontradiction.WeproceedasintheproofofLemma3.3.13writing q = q M + ::: + q 0 withhomogeneouspolynomials q j ofdegree j =0 ;:::;M: Expandthe product q asin.3.23.Thenweconcludethat k t q M =0and k t q M )]TJ/F28 7.9701 Tf 6.586 0 Td [(t + s + s q M =0.Thus,wecanwrite t q M = H M + t .3.28 t q M )]TJ/F28 7.9701 Tf 6.587 0 Td [(t + s + s q M = H M + s ; .3.29 where H M + t and H M + s arehomogeneous k -harmonicpolynomials.Nexttakethe innerproduct.3.27ofbothsidesofequation.3.29with q M .Then h q M )]TJ/F28 7.9701 Tf 6.586 0 Td [(t + s ;q M t i + h s ;q 2 M i = h H M + s ;q M i Usingequation3.3.28,wearriveat h q M )]TJ/F28 7.9701 Tf 6.587 0 Td [(t + s ;H M + t i + h s ;q 2 M i = h H M + s ;q M i : Now weuseTheorem3.3.14.Sincedeg H M + t > deg q M )]TJ/F28 7.9701 Tf 6.586 0 Td [(t + s +2 k )]TJ/F19 11.9552 Tf 12.519 0 Td [(1anddeg H M + s > deg q M +2 k )]TJ/F19 11.9552 Tf 11.915 0 Td [(1,thersttermontheleftandthetermontherightarebothzero. Thus, h s ;q 2 M i =0implies q M =0since 6 =0isnon-negative,acontradiction. 88

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4ValenceofHarmonicMapsandGravitationalLensing 4.1FixedPointsofConjugatedBlaschkeProductswithApplicationsto GravitationalLensing Thissectionistakenfromthejointpaper[71]withLudwigKuzniapublishedin ComputationalMethodsandFunctionTheory". Aconjectureinastronomywasrecentlyresolvedasanaccidentalcorollary ofatheoremregardingzerosofcertainplanarharmonicmaps.Asasteptoward extendingthefundamentaltheoremofalgebra,thetheoremgaveaboundof5 n )]TJ/F19 11.9552 Tf 12.173 0 Td [(5 forthenumberofzerosofafunctionoftheform r z )]TJ/F19 11.9552 Tf 12.99 0 Td [( z ,where r z isrationalof degree n .Inthissection,wewillinvestigatethecasewhen r z isaBlaschkeproduct. Theresultingsharpboundis n +3andtheproofissimple.Wediscussanapplication togravitationallensesconsistingofcollinearpointmasses. ThestrongesttestpassedbyEinstein'sgeneralrelativitywastheprediction ofthedeectionofstarlightfamouslyconrmedbyEddingtonduringasolareclipse [29].Besidesperturbingormagnifyinglightfromadistantstar,agravitationaleld cancreatemultipleimagesorevenanellipticalringfromasinglesourcesee[35]for earlyspeculationsmadebyEinsteinhimself.Inordertomodelthisphenomenon, wewillassumethattheso-calledgravitationallensconsistsentirelyofpoint-masses residinginacommonplaneperpendiculartoourlineofsightmodestviolationsof thisassumptionarenotsevere{wecanprojecttheoutlierstothelensplane.Thepath ofalightrayundertheinuenceofgravityfollowsgeodesicsofaspace-timemetric whichinturnisfoundbysolvingasystemofnonlinearPDE's.Asanexception, 89

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thespace-timemetricarisingfromasinglestarcanbecalculatedexactlyseethe discussionontheSchwarzschildsolutionin[40]andleadstotheEinsteindeection Figure4.1:Lenswithonemass. angle, =4 MG=r ,measuredbetweenentryand exitasymptotesofapassingphotonintermsof themass, M ,anddistance, r = j j ,atthepoint ofclosestapproach.Thisistheonlyresultwe requirefromgeneralrelativity.Basicgeometry doestherest. Considerrstasinglepointmassatthe originofthelensplane.Supposeastarwithposition w inthesourceplaneemitsalightraythat entersthelensplaneandisdeectedtowardour telescope.Thesmallangleapproximationgives s forthelengthofthevector, v ,whichhasthe samedirectionas .Thus, v =4 MG j j 2 .Nowthesimilarityofthetworighttriangles inFigure4.1leadstotherelationship w +4 sMG j j 2 = l + s l .Write z = l + s l and useunitswhichsubsumetheresultingconstantinfrontof M z j z j 2 .Thenthelensing equationforapointmassis z = w + M z j z j 2 .1.1 Nowsupposethelensplanecontains n deectorswithpositions z i andmasses m i .Iftheinteractionsamongpointmassescontrivingthelensareweakenough indicatingthatnonlineartermsintheeldequationsarenegligible,wecanfollow ourtemptationtotakethesuperpositionoftheEinsteindeectionanglesdueto individualmasses.Thelensingequationthenbecomes z = w + n X i =1 m i z )]TJ/F27 11.9552 Tf 11.955 0 Td [(z i j z )]TJ/F27 11.9552 Tf 11.955 0 Td [(z i j 2 .1.2 Thereplacement z )]TJ/F28 7.9701 Tf 6.586 0 Td [(z i j z )]TJ/F28 7.9701 Tf 6.587 0 Td [(z i j 2 = 1 z )]TJ/F25 7.9701 Tf 8.02 0 Td [( z i invitesacomplex-variablepointofview: 90

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z = w + n X i =1 m i z )]TJ/F19 11.9552 Tf 13.438 0 Td [( z i .1.3 Althoughitcanbeshownmathematicallythatsuchacongurationcanhave atmost n 2 +1images,in1997Mao,Petters,andWitt[82]suggestedtheboundwas actuallylinearin n .Rhierenedthisin2001[87],conjecturingthatagravitational lensconsistingof n pointmassescannotcreatemorethan5 n )]TJ/F19 11.9552 Tf 12.293 0 Td [(5imagesofagiven source.In2003,sheconstructedpointmasscongurationsforwhichthisboundis attained[88].Oneyearlater,KhavinsonandNeumann[60]provedaboundof5 n )]TJ/F19 11.9552 Tf 11.144 0 Td [(5 zerosforharmonicmappingsoftheform r z )]TJ/F19 11.9552 Tf 11.923 0 Td [( z ,where r z isrationalofdeg n> 1. Noticethatconjugatingbothsidesof.1.3putsitintheform z = r z .See[61] fortheexpositionandfurtherdetails. Inthenextsectionwewillconsideracasewhen5 n )]TJ/F19 11.9552 Tf 10.746 0 Td [(5isnotthebestpossible. Namely,werequirethat r z = B z isaniteBlaschkeproduct.Wewillprovea sharpboundforthiscaseusingintroductory-levelcomplexvariables.Thiswillnot immediatelygiveanyinsightintogravitationallensing,though,becauseallresidues of r z mustberealandpositiveinorderfor z = r z tocoincidewithalensing equation.ThisalmostneverhappensinthecaseofBlaschkeproducts.Inthethird section,webringaclassofphysicalexamplesintothepictureusingafamiliarM 'obius transformation. 4.1.1Case r z = B z Throughoutthispaper,weassumethatanyBlaschkeproductisnontriviali.e. B z 6 = z .Thegoalofthissectionistoshowthatmapsoftheform B z )]TJ/F19 11.9552 Tf 13.736 0 Td [( z canhaveatmost n +3zeros,where B z isaniteBlaschkeproduct.Recallthat aniteBlaschkeproductisafunctionoftheform Q n i =1 z )]TJ/F28 7.9701 Tf 6.586 0 Td [(a i 1 )]TJ/F25 7.9701 Tf 8.299 0 Td [( a i z ,where j a i j < 1for i =1 ;:::;n .First,wenoticethatsolutionsto B z = z aresymmetricwithrespect totheunitcircle.Thisfactisvariedviasimplealgebra. 91

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Lemma4.1.1 Let B z beaBlaschkeproductand z 0 2 D ,then B z 0 = z 0 ifand onlyif B = z 0 =1 =z 0 ,where 1 = 0= 1 bytheusualconvention. Thenextlemma,whoseproofisastandardexerciseinagraduatecourseof complexanalysisseeforexample[43],page265,givessomeavoroftheimportance ofBlaschkeproducts,statingthattheyarethe only analytic,boundary-preserving, self-mapsofthedisc.Onerecognizesthisasatrivialcaseofthefactorizationtheorem from H p theory. Lemma4.1.2 Suppose f z isananalyticmapof D intoitselfcontinuousuptothe boundarywhichsendstheboundarytotheboundary,then f z hasnitelymanyzeros in D .Furthermore, f isaBlaschkeproductwith n factors,where n isthenumberof zerosof f z in D Proof. Werstshowthat f hasnitelymanyzerosin D .Since f iscontinuousin D andmapstheboundarytotheboundary,thereexistsan r< 1suchthat f hasno zerosintheannulus f z : r< j z j 1 g .If f hadinnitelymanyzerosin D ,thenthey wouldallbein f z : j z j r g .However,thiswouldimplythat f 0.Hence f has nitelymanyzeros.NowformaBlaschkeproduct, B z = Q n i =1 z )]TJ/F28 7.9701 Tf 6.587 0 Td [(a i 1 )]TJ/F25 7.9701 Tf 8.299 0 Td [( a i z ,usingthezeros, a i ,of f z whichliein D .Noticethat g z = f z B z isanalyticandnonvanishingin D Therefore, u z =Re f log g z g =log j g z j isharmonicthroughout D .Moreover, on @ D j g z j = j f z j j B z j =1sothat u z j @ D =0.Itthenfollowsfromthemaximum principlethat u z isidenticallyzero.Therefore,log g z ,asapurelyimaginary, analyticfunction,mustbeconstantconsidertheCauchy-Riemannequations.Hence, f z isaunimodularmultiplei.e.arotationof B z Lemma4.1.3 If B z isaBlaschkeproduct,then B z = z hasatmostonesolution in D Proof. Supposethat z 0 2 D isaxedpointof B .Let z bethediscautomorphism z 0 )]TJ/F28 7.9701 Tf 6.586 0 Td [(z 1 )]TJETq1 0 0 1 121.874 104.873 cm[]0 d 0 J 0.359 w 0 0 m 8.09 0 l SQBT/F28 7.9701 Tf 121.874 100.186 Td [(z 0 z andset f = )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 B ,then f : D D and f =0.Thus,bytheSchwarz 92

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lemma,if f hasanotherxedpoint,then f z z .Since f isnottheidentitymap andxedpointsof B correspondtoxedpointsof f ,itfollowsthat B canhaveat mostonexedpointin D Nowweproveourmaintheorem. Theorem4.1.4 Let B z beaBlaschkeproductwith n factors;if B z 6 = z then B z = z hasatmost n +3 solutionsin C .Inparticular,thereare n +1 solutions on @ D ,andthereisasolutionin D ifandonlyifthereisasolutionin C )]TJETq1 0 0 1 500.319 554.235 cm[]0 d 0 J 0.478 w 0 0 m 8.634 0 l SQBT/F41 11.9552 Tf 500.319 544.325 Td [(D Proof. Webeginbyshowingthat B z = z has n +1solutionson @ D .Noticethat z =1 =z if z 2 @ D ,thus B z = z becomes B z =1 =z .Therefore,itisequivalent tosolve zB z =1for z 2 @ D .Now zB z isaBlaschkeproductwith n +1factors, henceitisan n +1foldcoveringof D withoutramicationpointsover @ D .Therefore, zB z =1has n +1distinctsolutionson @ D .Nextweinvestigatetheinterior.Notice thatif z 0 issuchthat B z 0 = z 0 ,then B B z 0 = z 0 .Thisleadsustoconsider B B z = z .Wenotethat B B z isarationalfunction,analyticin D ,thatmaps D toitselfandhasmodulusoneontheboundary.Hence,byLemma4.1.2, B B z isaBlaschkeproduct.ByLemma4.1.3, B B z = z hasatmostonesolutionin D andhence B z = z hasatmostonesolutionin D .Moreover,byLemma4.1.1,there isasolutionin C )]TJETq1 0 0 1 212.575 299.191 cm[]0 d 0 J 0.478 w 0 0 m 8.634 0 l SQBT/F41 11.9552 Tf 212.575 289.282 Td [(D ifandonlyifthereisasolutionin D ItisevidentfromtheproofofTheorem4.1.4that B z = z willalwayshave n +1solutionson @ D .Interestingly,thereareexampleshaving n +1solutionsand thosehaving n +3solutions.Forexample, zA z = z ,where A z isaBlaschke productwith n )]TJ/F19 11.9552 Tf 13.034 0 Td [(1factors,willhaveasolutionat z =0andthecorresponding solutionin C )]TJETq1 0 0 1 188.237 165.692 cm[]0 d 0 J 0.478 w 0 0 m 8.634 0 l SQBT/F41 11.9552 Tf 188.237 155.782 Td [(D z = 1 .Wepostponeanexamplewithexactly n +1solutionsuntil later. WenowmoveontoacorollaryofTheorem4.1.4. 93

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Corollary4.1.5 Let beasimplyconnectedJordandomain.Suppose f isanantianalyticmapof toitselfwhichmaps @ toitself f isproper".Then f w )]TJ/F27 11.9552 Tf 12.133 0 Td [(w hasatmost n +2 zerosin ,where n isthedegreeof f w asaself-mapof @ i.e. foreach w 2 @ f )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 w has n elements. Proof. Let betheRiemannmapofonto D .Then f )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 z isananalytic, boundary-preservingself-mapof D with n zerosbytheargumentprinciple.By Lemma4.1.2, f )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 z = B z ,where B z isaBlaschkeproductwith n factors. Thus, f )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 z = B z whichhasatmost n +2xedpointsin D byTheorem 4.1.4.Wehave f )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 z 0 = z 0 i f w 0 = w 0 ,where w 0 = )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 z 0 .Thus, f w hasatmost n +2xedpointsin 4.1.2GravitationalLensingbyCollinearPointMasses Supposethepositionsofthemasses z i in.1.3alongwithprojectionofthesource w arecollinear.Thenwithoutlossofgeneralitywemayassumetheyareonthereal axis.Thenthelensingmapsendsthereallinetoitself,theupperhalfplanetoitself, andthelowerhalfplanetoitself.AsintheproofofCorollary4.1.5,weconjugate f z = w + P n i =1 m i z )]TJ/F25 7.9701 Tf 8.019 0 Td [( z i withtheMobiustransformation, z = z )]TJ/F28 7.9701 Tf 6.587 0 Td [(i z + i ,whichsendsthe upperhalfplaneto D .Itisinfacttruethatallrationalfunctionscorresponding toniteBlaschkeproductshavetheform bz + w + P n i =1 m i z i )]TJ/F28 7.9701 Tf 6.586 0 Td [(z ,where m i w ,and z i areasaboveand b isnonnegative,andthedegreeoftheBlaschkeproductis n if b =0and n +1if b 6 =0.Foramoreinformationonthismoregeneralresultonthe correspondencebetweenBlaschkeproductsandrationalfunctionssee[92].Butin thiscase, anditsinversearedenedintheentireplane,sowegetatotalestimate ratherthanjustintheupperhalfplaneofatmost n +3solutionsofthelensing equation.Wesummarizethisas Corollary4.1.6 Thereareatmost n +3 imageslensedbyacollinearconguration ofpointmasseswhentheprojectionofthesourceontothelensplaneisalsocollinear. 94

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Figure4.2:Dotsrepresentmassesdeectors,circlesrepresentlensedimagesandtheXat theoriginistheprojectionofthesource. The n +1solutionslocatedontherealaxisarenotsurprising.Indeed,the n massesdividethereallineinto n +1intervals.Considerforinstanceaniteinterval betweentwomasses.Ifarayfromthesourcelandstooclosetotheleftendpoint itwillbedeectedtoosharplyinthatdirectionandmissourtelescope.Iftheray landstooclosetotherightendpoint,itwillbedeectedtoosharplyintheother direction.Weexpectanintermediatevaluewheretherayisproperlydeected.Also notsurprisingisthesymmetryofthetwoimagesthatoccurotherealaxis.But whatisnotphysicallyobviousisthatthereshouldonlybetwosuchimages. Foranexample,weconsiderthefollowinglensingequation z = 3 z )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 + 4 z +4 + 1 z +1 : .1.4 Herewehavemassesof3 ; 4,and1atthepoints1 ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(4,and )]TJ/F19 11.9552 Tf 9.298 0 Td [(1,respectivelyandthe observer,originofthelensplaneandthesourcearecollinear.Asmentionedabove,it isexpectedtohavefourimagesontherealaxis,whichareseparatedbythemasses. Inthiscase,theapproximatelocationsoftheseimagesare2 : 63 ; )]TJ/F19 11.9552 Tf 9.299 0 Td [(0 : 18 ; )]TJ/F19 11.9552 Tf 9.298 0 Td [(1 : 51,and )]TJ/F19 11.9552 Tf 9.299 0 Td [(4 : 97.Wealsohavethetwosymmetricimagesotherealaxis,whicharelocatedat approximately0 : 78 i 2 : 01.SeeFigure4.2foradepictionofthisexample. 95

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WenowreturntothetaskofndingaBlaschkeproduct, B ,suchthat B z = z hasexactly n +1solutionsin C .Todothis,weexamineanotherlensingequation, namely z = 1 z )]TJ/F19 11.9552 Tf 11.955 0 Td [(1 + 10 z +1 : .1.5 Rearranging.2,wehave z )-222(j z j 2 =9 )]TJ/F19 11.9552 Tf 11.955 0 Td [(2Re z 4.1.6 Noticethatsolutionsto.3mustberealorlieonthecircle f z : j z j = p 10 g .Since 0=9 )]TJ/F19 11.9552 Tf 12.063 0 Td [(2 p 10cos hasnosolutions,wemayconcludethat.3hasnosolutionson f z : j z j = p 10 g .Therefore,anysolutionsto.3mustbereal.Fromthiswemay concludethat.2hasexactly3solutionsin C .Afterconjugatingbytheappropriate Mobiustransformation,weobtainaBlaschkeproduct, B ,suchthat B has2factors and B z = z has3solutions. Remarks :1.IfthehypothesisofCorollary4.1.5couldbeweakened,itwouldhave potentialasatoolforlocallyanalyzinglensingmapsinregionswhicharesentto themselves. 2.Asidefromaboundof n 2 providedbyanargumentofWilmshurst[106]along withBezout'sTheorem,littleprogresshasbeenmadeboundingzerosofharmonic polynomials p z + q z whendeg p z > deg q z > 1Thequestionregardinga boundonthenumberofzerosofharmonicpolynomialswasraisedbyT.Sheil-Small [98].Perhapsboundingsolutionsof B z = A z ,with B z and A z Blaschke productscouldprovelessstubborn. 3.Itisofinteresttoaskhowsensitiveimagesaretosmallperturbationsofthe positionofoneofthelensingmasses.Thiscorrespondstoagravitationallenswhich includesastarpossessingaplanet.Inpractice,microlensingtechniquessearchfora changeinthe brightness ofalensedimageinordertodetectaplanet.Interestingly, inthecollinearsystemsconsideredinthispaper,the position ofthesymmetricpair ofimagescanbeespeciallysensitivetoperturbationsofthemasses.Inorderto 96

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deemthisanalternativetechnique,aninformedinvestigationusingrealisticmassand distancescaleswouldbenecessary. 4.2TranscendentalHarmonicMappingsandGravitationalLensingby IsothermalGalaxies Thissectionistakenfromthejointpaper[58]withDmitryKhavinsonpublishedin ComplexAnalysisandOperatorTheory. UsingtheSchwarzfunctionofanellipse,itwasrecentlyshownthatgalaxies withdensityconstantonconfocalellipsescanproduceatmostfourbright"images ofasinglesource.Themorephysicallyinterestingexampleofanisothermalgalaxy hasdensitythatisconstanton homothetic ellipses.Inthatcasebrightimagescanbe seentocorrespondtozerosofacertaintranscendentalharmonicmapping. 4.2.1Asimpleproblemincomplexanalysiswithadirectapplication Wewillusecomplexdynamicstogiveanupperboundonthetotalnumberofsolutions oftheequation arcsin k z + w = z; .2.7 where w isacomplexparameter,and k isarealparameter. Ourmotivationfordoingsoisthatsolutionsof.2.7infactcorrespondtovirtualimagesobservedwhenthelightfromadistantsourcepassesnearanisothermal, ellipsoidalgalaxy.Indeed,usingthecomplexformulationofthethin-lensapproximation[99],thelensingequationiscalculatedbyndingtheCauchytransformofthe massdistributionprojectedtothelensplane".Thiswascarriedoutin[38]withthe followingresultforthelensingequationofanisothermalgalaxy. C arcsin c + = .2.8 97

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WesketchthederivationofEq..2.8attheendofthissectionforthereader's convenience.Here,wetaketheprincipalbranchofarcsin, C and c arerealconstants dependingontheellipticalprojectionofthegalaxyontothelensplane,and isthe positionofthesourceprojectedtothelensplane.Valuesof whichsatisfy.2.8 givepositionsoftheobservedimages.Changingvariablesto z = )]TJ/F28 7.9701 Tf 6.587 0 Td [(! C w = != C ,and k = c= C puts.2.8intotheformofequation.2.7whilepreservingthenumberof solutions. Weshouldmentionthattheanti-analyticpotentialinthelensingequationconsideredhereandalsoin[18]and[38]diersfromthepotentialinthelensingequation inthemodeloftenusedbyastrophysicistssee[54]andthereferencestherein,where theprojectedmassdensityissupportedintheentirecomplexplane.Bothmodels usetheisothermal"densityproportionalto1 =t onellipses f x 2 =a 2 + y 2 =b 2 = t 2 g a and b xed.Themodelconsideredherethatyieldsequation.2.8assumesthat thedensityiszeroforall t greaterthansomevalueseeendofthissection.Letting thedensityhaveinnitesupportassumesthatthegalaxyhasinnitemassandlls theuniverse,yetitisthesimplestwaytoavoidgivingthegalaxyasharpedge", andastronomershavefoundthatthemodelbehavesreasonablyintheregionwhere thelensedimagesoccur.Weconsiderthemodelwithphysicallyrealisticcompact supportbutlessrealisticsharpedge"foramathematicalreason:inthatsetting, lensedimagesdescribedbysolutionsofequation.2.7correspondtozerosofa harmonic function.Wenotethatmodelswithsharpedges"havebeenconsideredby astrophysicistsaswell,cf.therecentpreprints[83]and[84]. Forgravitationallensesconsistingof n pointmasses,Mao,Petters,andWitt [82]suggestedthattheboundforthenumberofimageswaslinearin n .Bezout's theoremprovidesaboundquadraticin n .Rhierenedthisin2001,conjecturing thatagravitationallensconsistingof n pointmassescannotcreatemorethan5 n )]TJ/F19 11.9552 Tf 11.437 0 Td [(5 imagesofagivensource[87].In2003,sheconstructedpoint-masscongurationsfor whichtheseboundsareattained[88].D.KhavinsonandG.Neumann[60]settled herconjecturebygivingaboundof5 n )]TJ/F19 11.9552 Tf 11.998 0 Td [(5zerosforharmonicmappingsoftheform 98

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r z )]TJ/F19 11.9552 Tf 13.483 0 Td [( z ,where r z isrationalofdegree n> 1.See[60]fortheexpositionand furtherdetails.Solutionsof.2.7arezerosofa transcendental harmonicfunction, soextendingthetechniquesusedin[60]willrequiresomecareapriori,itisnoteven clearthatthenumberofzerosisnite,cf.[13],[69].Still,ourapproachdrawsonthe sametwomainresults:itheargumentprinciplegeneralizedtoharmonicmappings andiitheFatoutheoremfromcomplexdynamicsregardingtheattractionofcritical points.Inthenextsection,wewillformulatei.iiwillhavetobemodiedforour purposes,soideasfromcomplexdynamicsareworkedfromscratchintotheproofof Lemma4.2.5. 4.2.2Preliminaries:TheArgumentPrinciple Inordertostatethegeneralizedargumentprinciplesee[27]foracompleteexposition andproof,weneedtodenetheorderofazeroorpoleofaharmonicfunction.A harmonicfunction h = f + g ,where f and g areanalyticfunctions,iscalled sensepreserving at z 0 iftheJacobian Jh z = j f 0 z j 2 )-264(j g 0 z j 2 > 0forevery z insome puncturedneighborhoodof z 0 .Wealsosaythat h is sense-reversing if h issensepreservingat z 0 .If h isneithersense-preservingnorsense-reversingat z 0 ,then z 0 is calledsingularandnecessarilybutnotsuciently Jh z 0 =0,cf.[27],Ch.2. Thenotation C arg h z denotestheincrementintheargumentof h z along acurve C .The order ofanon-singularzeroisgivenby 1 2 C arg h z ,where C isa sucientlysmallcirclearoundthezero.Theorderispositiveif h issense-preserving atthezeroandnegativeif h issense-reversing.Suppose h isharmonicinapunctured neighborhoodof z 0 .Wewillreferto z 0 asapoleof h if h z !1 as z z 0 .Following [102],the order ofapoleof h isgivenby )]TJ/F25 7.9701 Tf 13.069 4.708 Td [(1 2 C arg h z ,where C isasuciently smallcirclearoundthepole.Wenotethatif h issense-reversinginsomepunctured neighborhoodofthepole,thentheorderofthepolewillbenegative.Wewillusethe followingversionoftheargumentprinciplewhichistakenfrom[102]: Theorem4.2.1 Let F beharmonic,exceptforanitenumberofpoles,inaJordan domain D .Let C beaclosedJordancurvecontainedin D notpassingthroughapole 99

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orazero,andlet R betheopen,boundedregionsurroundedby C .Suppose F hasno singularzerosin R andlet N bethesumoftheordersofthezerosof F in R .Let P bethesumoftheordersofthepolesof F in R .Then C arg F z =2 N )]TJ/F27 11.9552 Tf 11.955 0 Td [(P 4.2.3AnUpperBoundfortheNumberofImages Lemma4.2.2 Thesolutionsofequation.2.7areallcontainedintherectangle R := fj Re z j = 2 ; j Im z j M g ,where M issucientlylarge. Proof. Therequirementthat j Re z j = 2isimmediatesincethisstripistheimage of C undertheprincipalbranchofarcsin.Toseethatthereexistsan M suchthat solutionsof.2.7satisfy j Im z j M ,takesinofbothsides.Thisleadsto k sin z = z + w: .2.9 Weconsiderthemodulusofeachsideof.2.9for z = x + iy withlarge valuesof j y j .Recall,sin x + iy =sin x cosh y + i cos x sinh y .As y j k sin x + iy j = k= p sin 2 x cosh 2 y +cos 2 x sinh 2 y 0,uniformlyin x .Ontheother hand, j z + w j!1 Remark: Bythislemma,wecanboundthenumberofsolutionsof.2.7byboundingthenumberofzerosof F z := z + w )]TJ/F28 7.9701 Tf 22.163 4.707 Td [(k sin z intherectangle, R .Letuscalculatetheincrementoftheargumentof F z when @R istracedcounterclockwise. If @R passesthroughazeroof F thenwecaninsteadconsidertheboundaryof R := fj Re z j = 2+ "; j Im z j M g inthefollowingLemmaandintherestof thissection.Withasmallchoiceof ,thecalculationinthefollowingproofdoesnot change. Lemma4.2.3 @R arg F z )]TJ/F19 11.9552 Tf 21.918 0 Td [(2 ,where F z := z + w )]TJ/F28 7.9701 Tf 21.47 4.707 Td [(k sin z 100

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Proof. Considerthefourlinks V t = 2 i )]TJ/F27 11.9552 Tf 9.299 0 Td [(M + t ,0 t 2 M H t = 2 )]TJ/F27 11.9552 Tf 13.171 0 Td [(t iM ,0 t ,whichtracetheright,left,top,andbottomedges, respectively.Weneedtodeterminetheeectoftheterm )]TJ/F28 7.9701 Tf 18.813 4.707 Td [(k sin z .Withoutthisterm, F z isjustthetranslation z 7! z + w ,andinthatcase F V t and F H t trace theedgesofthetranslatedrectangle. Bychoosing M largeenoughinthepreviouslemma,wecanneglecttheterm )]TJ/F28 7.9701 Tf 18.813 4.707 Td [(k sin z onthetopandbottomedges.Ontherightedge, k sin V + t = k cosh )]TJ/F28 7.9701 Tf 6.586 0 Td [(M + t is purerealandincreasesmonotonicallyfromasmallvalueat t =0tothevalue k at t = M .Ontheinterval M t 2 M k cosh )]TJ/F28 7.9701 Tf 6.586 0 Td [(M + t decreases monotonicallyfrom k at t = M backtotheoriginalvalueat t =2 M .Similarly,ontheleftedge, k sin V )]TJ/F25 7.9701 Tf 6.254 1.073 Td [( t = )]TJ/F28 7.9701 Tf 32.104 4.707 Td [(k cosh )]TJ/F28 7.9701 Tf 6.587 0 Td [(M + t > )]TJ/F27 11.9552 Tf 9.298 0 Td [(k .Thus,theeectoftheterm )]TJ/F28 7.9701 Tf 18.813 4.707 Td [(k sin z istobendtheleft andrightsidesofthetranslatedrectangleinward,sothattheycrosseachotherifand onlyif k> 2 comparethetwoimagesingure4.3. If0 = 2,thentheimagesoftheleftandrightedgesintersectexactlytwice. Inthiscase,thereisathirdpossibilityinwhich @R arg F z = )]TJ/F19 11.9552 Tf 9.299 0 Td [(2 .Seetheright imageingure4.3. Denethefunction f z := k sin z )]TJ/F19 11.9552 Tf 15.181 0 Td [( w ,andnoticethatxedpointsof f z coincidewiththezerosof F z .Also,denethefunction f # z = k sin z )]TJ/F27 11.9552 Tf 13.092 0 Td [(w so that f # z = f z ,anddenotethecomposition f # f z by g z .Suppose z 0 isa zeroof F z .Then g z 0 = f # z 0 = f z 0 = z 0 ,sothat z 0 isaxedpointofthe analyticfunction g z .Moreover,if z 0 isasense-preservingzerothen j f 0 z 0 j < 1and g 0 z 0 = f # 0 z 0 f 0 z 0 = j f 0 z 0 j 2 < 1,sothat z 0 isan attracting xedpointof g see [28]and[21].Finally,if z 0 isasingularzerothen g 0 z 0 =1notjustinmodulus! 101

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Figure4.3:Theimageof @R under F z with w =0andtwochoicesforthevalueof k .In theleft, k =1 <= 2.Inthiscase, 1 2 @R arg F z =1.Ifweset w to,say,1thenwehave 1 2 @R arg F z =0.Intheright, k =2 >= 2,and 1 2 @R arg F z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Ifweset w to, say,1or i thenwehave 1 2 @R arg F z =0or1,respectively. sothat z 0 isaso-called parabolic xedpointof g Weusecomplexdynamicstoboundthenumberofattractingandparabolic xedpointsof g z .TheversionoftheFatoutheoremfoundinmosttextbookson complexdynamicssuchas[21]boundsthenumberofattractingxedpointsofa polynomialorrationalfunctionbythenumberofcriticalpoints.Thisfallsshort, sinceweareconsideringhereafunction g withinnitelymanyessentialsingularities andinnitelymanycriticalpoints.However,themoreupdatedversionoftheFatou TheoremprovidedbythefollowingLemmafoundin[11]weformulateaspecialcase ofLemma10inthatpaperisperfectlysuitedtooursituation.Foranexpositionof theextensionsoftheFatoutheoremleadinguptothismodernformulation,seethe survey[17]. Lemma4.2.4 Suppose g ismeromorphicoutsideanatmostcountablyinnite,compactsetconsideredasasubsetoftheRiemannsphere ^ C := C [f1g ofessential 102

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singularities.Thenthebasinofattractionofanattractingorparabolicxedpoint z 0 containstheforwardorbitofsomesingularpointof g )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 Lemma4.2.5 Thenumberofsense-preservingzeros, n + ,of F plusthenumberof singularzeros, n 0 ,isatmost3. Proof. Suppose z 0 isasense-preservingorsingularzeroof F .Then,bythediscussion above, z 0 isanattractingorparabolicxedpointof g z = f # f z .Wenotethat g ismeromorphicexceptat z = 1 andatthecountablymanyzerosof f z converging to 1 ,sothat g satisesLemma4.2.4. Forthesetofsingularpointsof g )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 ,Sing g )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 ,wehave Sing g )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 =Sing f # )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 [ f # Sing f )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 : Wecanndexplicitly, f # )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 =arcsin k + w ,sothat Sing f # )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 = f)]TJ/F27 11.9552 Tf 15.276 0 Td [(w; )]TJ/F27 11.9552 Tf 9.298 0 Td [(w k g : Similarly,bywriting f )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 explicitlyweseethat f # Sing f )]TJ/F25 7.9701 Tf 6.587 0 Td [(1 = f f # )]TJ/F19 11.9552 Tf 11.693 0 Td [( w ;f # )]TJ/F19 11.9552 Tf 11.692 0 Td [( w k g ; soSing g )]TJ/F25 7.9701 Tf 6.586 0 Td [(1 isthesetofatmostsixpoints f)]TJ/F27 11.9552 Tf 15.276 0 Td [(w; )]TJ/F27 11.9552 Tf 9.299 0 Td [(w k;f # )]TJ/F19 11.9552 Tf 11.692 0 Td [( w ;f # )]TJ/F19 11.9552 Tf 11.692 0 Td [( w k g .By Lemma4.2.4, z 0 attractsatleastoneofthesepoints,givingusthebound n 0 + n + 6. Thefollowingobservationimprovesthisestimate. Let z c beoneofthethreepoints )]TJ/F27 11.9552 Tf 9.299 0 Td [(w; )]TJ/F27 11.9552 Tf 9.299 0 Td [(w k andsuppose z 0 attracts z c .We claimthat z 0 alsoattracts f # z c .Indeed, lim n !1 g n f # z c =lim n !1 f # f n f # z c =lim n !1 f g n z c = f z 0 = z 0 ; sothat g n f # z c convergesto z 0 .Thus,eachsense-preservingorsingularzeroof F attracts,underiterationof g ,oneofthethreepoints f # )]TJ/F19 11.9552 Tf 11.692 0 Td [( w f # )]TJ/F19 11.9552 Tf 11.692 0 Td [( w k .So 103

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n 0 + n + 3. Theorem4.2.6 Thenumberofsolutionsto.2.7isboundedby8. Proof. BytheremarkfollowingLemma4.2.2,thetotalnumberofsolutionsto.2.7 equalsthetotalnumberofzerosof F z in R .Recallthat F z iscalledregular" ifitisfreeofsingularzerossee[66]and[28].Supposeforthemomentthat F z isregularsothatTheorem4.2.3applies.Then,thetotalnumberofzerosof F z in R is n + + n )]TJ/F19 11.9552 Tf 7.084 1.793 Td [(,where n + and n )]TJ/F19 11.9552 Tf 11.383 1.793 Td [(count,respectively,thesense-preservingandsensereversingzerosof F z in R .ByLemma4.2.3andTheorem4.2.1, )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 N )]TJ/F27 11.9552 Tf 10.927 0 Td [(P F z hasonesense-reversingpolein R oforder )]TJ/F19 11.9552 Tf 9.299 0 Td [(1andallnon-singularzerosareoforder 1.ByLemma4.2.5 )]TJ/F19 11.9552 Tf 9.299 0 Td [(1 3 )]TJ/F27 11.9552 Tf 11.955 0 Td [(n )]TJ/F19 11.9552 Tf 9.741 1.793 Td [(+1,sothat n )]TJ/F20 11.9552 Tf 10.406 1.793 Td [( 5.Thus, n + + n )]TJ/F20 11.9552 Tf 10.405 1.793 Td [( 8. Fix k .Thereisadensesetofparameters w forwhich F z isregular.Indeed, considertheimageof f z : j d dz k sin z j =1 g under z )]TJ/F28 7.9701 Tf 19.853 4.707 Td [(k sin z .Thissethasemptyinterior, andif w isinitscomplement, F z isfreeofsingularzeros. Nowsuppose F z isnotregular.ThenLemma4.2.5stillappliessothat n 0 + n + 3,butthepreviousargumentforbounding n )]TJ/F19 11.9552 Tf 9.875 1.794 Td [(doesnot.If F z isperturbed byasucientlysmallconstanttoobtain F z ,thenumberofsense-reversingzeros doesnotdecreasebycontinuityoftheargumentprincipleinasense-reversingregion. Thezerossimplymoveinasmallneighborhoodofeachsense-reversingzero.Bythe preceding,wecanchooseaperturbation F z thatisregularandthereforehasat mostvesense-reversingzeros.Thisgives n )]TJ/F20 11.9552 Tf 10.405 1.793 Td [( 5,and n 0 + n + + n )]TJ/F20 11.9552 Tf 10.406 1.793 Td [( 8. 4.2.4Remarks Sofar,astronomershaveonlyobservedupto5imagesbright+1dimproducedby anellipticallensseegure4.4.In[54]therehavebeenconstructedexplicitmodels dependingonthesemiaxesoftheellipsehaving9imagesbright+1dimbutonly inthepresenceofashear,i.e.alineargravitationalpullfrominnityaterm z addedtoequation.2.7.Sofar,wehavenotbeenabletoobtainauniversalbound 104

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Figure4.4:Fourimagesofalightsourcebehindanellipticalgalaxy.Credit:NASA,Kavan Ratnatunga,JohnsHopkinsUniv. inthepresenceofashearthatissimilartoTheorem4.2.6.Itseemed,basedonNASA observations,naturaltoconjecturethat,intheabsenceofshear,therecanbeatmost 4brightimages.Yet,recentlyW.BergweilerandA.EremenkoimprovedTheorem 4.2.6byshowingthatthereareatmomst6brightimages,andtheygeneratedan examplewith6brightimages[18].Withtheirkindpermission,weincludetheir exampleseegure4.5.Forthecasewithshear,weconjecturethefollowing. Conjecture4.2.7 Thenumberofbrightimageslensedbyanisothermalelliptical galaxywithcompactlysupportedmassdensitywithshearisatmost8. Wecautionthereaderthatin[54]themassdensitywasassumedtobeextended allthewaytoinnity,sothelensingpotentialin[54]wasdierentfromtheonewe considerhereandin[38],[18]. 4.2.5Derivationofthecomplexlensingequationfortheisothermalellipticalgalaxy Supposethatlightfromadistantsourcestarisdistortedasitpassesbyanintermediate,continuousdistributionofmasswhichdoesnotdeviatetoofarfrombeing containedinacommonplanethelensplane"perpendiculartoourlineofsight. Let z denotetheprojectedmassdensity.ThenbasicresultsfromGeneralRelativitycombinedwithGeometricOpticssee[99]leadtothefollowinglensingequation 105

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Figure4.5:Equation.2.7has6solutionswhen k =1 : 92and w = )]TJ/F49 10.9091 Tf 8.485 0 Td [(: 67 i .Choosing a =1 ;b = : 041,and M =2seebelowleadsto k =1 : 92andgivesthepictureofthesix imagesshownhereinthe -planealongwiththegalaxy'sellipticalsilhouetteandthesource plottedasbox. relatingthepositionofthesourceprojectedtothelensplane w tothepositionsof lensedimages z z = Z dA )]TJ/F19 11.9552 Tf 12.664 0 Td [( z + w .2.10 Consider,rst,thecasewhentheprojecteddensity z = D isconstantand supportedon:= f x 2 a 2 + y 2 b 2 1 ;a>b> 0 g ,anellipse.Thenequation.2.10 becomes z = Z DdA )]TJ/F19 11.9552 Tf 12.664 0 Td [( z + w: BythecomplexGreen'sformula,for z outsidei.e.,forbright"images, thisbecomes z = D 2 i Z @ d )]TJ/F19 11.9552 Tf 12.664 0 Td [( z + w: 106

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TheSchwarzfunctionbydenition,analyticand= on @ fortheellipse equals c 2 = a 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(b 2 : S = a 2 + b 2 c 2 )]TJ/F19 11.9552 Tf 13.151 8.088 Td [(2 ab c 2 p 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(c 2 = a 2 + b 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(2 ab c 2 + 2 ab c 2 )]TJ/F32 11.9552 Tf 11.955 10.949 Td [(p 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(c 2 = S 1 + S 2 where S 1 isanalyticin ,and S 2 isanalyticoutsideand S 2 1 =0.Since z is outside,combiningthiswithCauchy'sformulagives 1 2 i Z @ S d )]TJ/F19 11.9552 Tf 12.664 0 Td [( z + w = 2 ab c 2 D z )]TJ 11.955 10.418 Td [(p z 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(c 2 + w fortheright-hand-sideofthelensingequation. Nextconsiderthecaseofisothermal"densitysupportedon, = M=t on @ t t := t = f x 2 a 2 + y 2 b 2 t 2 g ;t< 1,and M aconstant. ThentheCauchypotentialterminthelensingequation.2.10becomes Z z )]TJ/F19 11.9552 Tf 12.665 0 Td [( z dA = Z 1 0 M t d dt Z t dA )]TJ/F19 11.9552 Tf 12.665 0 Td [( z dt .2.11 Fortheinsideintegral,weseethat R t dA )]TJ/F25 7.9701 Tf 7.078 0 Td [( z = t 2 R dA t )]TJ/F25 7.9701 Tf 7.078 0 Td [( z = t R dA )]TJ/F25 7.9701 Tf 7.078 0 Td [( z=t which accordingtoourpreviouscalculationis C 0 z )]TJ 12.531 9.835 Td [(p z 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(c 2 t 2 ,wheretheconstant C 0 dependsonlyon.Nowthe t -derivativeofthisis C 0 t p z 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(c 2 t 2 .Thus.2.11becomes MC 0 R 1 0 dt p z 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(c 2 t 2 Finally,wearriveat.2.8,thelensingequationfortheisothermalelliptical galaxy, z = C arcsin c z + w; where C = 2 ab c M 107

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