Holder continuity of green's functions

Citation
Holder continuity of green's functions

Material Information

Title:
Holder continuity of green's functions
Creator:
Tookos, Ferenc
Place of Publication:
[Tampa, Fla.]
Publisher:
University of South Florida
Publication Date:
Language:
English

Subjects

Subjects / Keywords:
Logarithmic capacity
Newtonian potential
Equilibrium measure
Boundary behavior
Wiener's criterion
Dissertations, Academic -- Mathematics -- Doctoral -- USF ( lcsh )
Genre:
government publication (state, provincial, terriorial, dependent) ( marcgt )
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
ABSTRACT: We investigate local properties of the Green function of the complement of a compact set$E$. First we consider the case $E\subset [0,1]$ in the extended complex plane. We extend a result of V. Andrievskii which claims that if the Green function satisfies the H\"older-$1/2$ condition locally at the origin, then the density of $E$ at $0$, in terms of logarithmic capacity, is the same as that of the whole interval $[0,1]$. We give an integral estimate on the density in terms of the Green function, which also provides a necessary condition for the optimal smoothness. Then we extend the results to the case $Esubset [-1,1]. In this case the maximal smoothness of the Green function is "older-1 and a similar integral estimate and necessary condition hold as well.In the second part of the paper we consider the case when $E$ is acompact set in R, > 2. We give a Wiener type characterization for the "older continuity of the Green function, thus extending a result of L.
Thesis:
Thesis (Ph.D.)--University of South Florida, 2004.
Bibliography:
Includes bibliographical references.
System Details:
System requirements: World Wide Web browser and PDF reader.
System Details:
Mode of access: World Wide Web.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 65 pages.
General Note:
Includes vita.
Statement of Responsibility:
by Ferenc Tookos.

Record Information

Source Institution:
University of South Florida Library
Holding Location:
University of South Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
001680988 ( ALEPH )
62498641 ( OCLC )
E14-SFE0001137 ( USFLDC DOI )
e14.1137 ( USFLDC Handle )

Postcard Information

Format:
Book

Downloads

This item has the following downloads:


Full Text
xml version 1.0 encoding UTF-8 standalone no
record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd
leader nam Ka
controlfield tag 001 001680988
003 fts
005 20060215071058.0
006 m||||e|||d||||||||
007 cr mnu|||uuuuu
008 051207s2004 flu sbm s000 0 eng d
datafield ind1 8 ind2 024
subfield code a E14-SFE0001137
035
(OCoLC)62498641
SFE0001137
040
FHM
c FHM
049
FHMM
090
QA36 (Online)
1 100
Tookos, Ferenc.
0 245
Holder continuity of green's functions
h [electronic resource] /
by Ferenc Tookos.
260
[Tampa, Fla.] :
b University of South Florida,
2004.
502
Thesis (Ph.D.)--University of South Florida, 2004.
504
Includes bibliographical references.
516
Text (Electronic thesis) in PDF format.
538
System requirements: World Wide Web browser and PDF reader.
Mode of access: World Wide Web.
500
Title from PDF of title page.
Document formatted into pages; contains 65 pages.
Includes vita.
3 520
ABSTRACT: We investigate local properties of the Green function of the complement of a compact set$E$. First we consider the case $E\subset [0,1]$ in the extended complex plane. We extend a result of V. Andrievskii which claims that if the Green function satisfies the H\"older-$1/2$ condition locally at the origin, then the density of $E$ at $0$, in terms of logarithmic capacity, is the same as that of the whole interval $[0,1]$. We give an integral estimate on the density in terms of the Green function, which also provides a necessary condition for the optimal smoothness. Then we extend the results to the case $Esubset [-1,1]. In this case the maximal smoothness of the Green function is "older-1 and a similar integral estimate and necessary condition hold as well.In the second part of the paper we consider the case when $E$ is acompact set in R, > 2. We give a Wiener type characterization for the "older continuity of the Green function, thus extending a result of L.
590
Adviser: Vilmos Totik, Ph.D.
653
Logarithmic capacity.
Newtonian potential.
Equilibrium measure.
Boundary behavior.
Wiener's criterion.
690
Dissertations, Academic
z USF
x Mathematics
Doctoral.
773
t USF Electronic Theses and Dissertations.
4 856
u http://digital.lib.usf.edu/?e14.1137



PAGE 1

by FerencTookos Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofMathematics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:VilmosTotik,Ph.D. MouradE.H.Ismail,Ph.D. EvgueniiA.Rakhmanov,Ph.D. BorisShekhtman,Ph.D. DateofApproval: October1,2004 Keywords:logarithmiccapacity,Newtonianpotential,equilibriummeasure,boundarybehavior,Wiener'scriterion WorkwassupportedbyNSFDMS-0097484andNSFDMS-040650 cCopyright2005,FerencTookos

PAGE 3

Iextendmythankstoallthesupervisingcommitteemembers,MouradE.H.Is-mail,EvgueniiA.RakhmanovandBorisShekhtmanforcreatinganexcellentresearchatmosphereinApproximationTheory. IalsothankmyprofessorsattheUniversityofSzegedforprovidingmewithastrongeducation.Specialthanksareduetomyanalysisprofessors,L.Hatvani,J.Heged}us,L.LeindlerandL.Kerchy. Finally,Iowemuchgratitudetomyparentsandmybrotherfortheirunerringsupportfromafar.

PAGE 4

Abstractiii 1Introduction1 2OptimalSmoothnessforE[0;1]5 2.1Notations,Denitions...........................5 2.2Results...................................6 2.3ProofofTheorem2.2.1..........................8 2.4ProofofTheorem2.2.3..........................14 2.5Lemmas..................................18 3AWiener-typeConditioninRd29 3.1Preliminaries...............................29 3.2Results...................................34 3.3ProofofTheorem3.2.1..........................36 Proofofa)inTheorem3.2.1.......................36 Proofofb)inTheorem3.2.1......................40 3.4ProofofLemma3.2.2...........................54 References57 AbouttheAuthorEndPagei

PAGE 5

2.2Themappings 1and 2.........................24ii

PAGE 6

FerencTookos FirstweconsiderthecaseE[0;1]intheextendedcomplexplane.WeextendaresultofV.AndrievskiiwhichclaimsthatiftheGreenfunctionsatisestheHolder-1=2conditionlocallyattheorigin,thenthedensityofEat0,intermsoflogarithmiccapacity,isthesameasthatofthewholeinterval[0;1].WegiveanintegralestimateonthedensityintermsoftheGreenfunction,whichalsoprovidesanecessaryconditionfortheoptimalsmoothness.ThenweextendtheresultstothecaseE[)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1].InthiscasethemaximalsmoothnessoftheGreenfunctionisHolder-1andasimilarintegralestimateandnecessaryconditionholdaswell. InthesecondpartofthepaperweconsiderthecasewhenEisacompactsetinRd,d>2.WegiveaWienertypecharacterizationfortheHoldercontinuityoftheGreenfunction,thusextendingaresultofL.CarlesonandV.Totik.Theobtaineddensityconditionisnecessary,anditissucientaswell,providedEsatisestheconecondition.ItisalsoshownthattheHolderconditionfortheGreenfunctionataboundarypointcanbeequivalentlystatedintermsoftheequilibriummeasureandthesolutiontothecorrespondingDirichletproblem.Theresultssolvealongstandingopenproblem-raisedbyMaz'jainthe1960's-underthesimpleconecondition.iii

PAGE 7

SupposethatECisacompactsetwithpositivelogarithmiccapacitycap(E)>0.Let:= Supposethat0isaregularpointofE,i.e.,g(z)iscontinuousat0andg(0)=0.FirstconsiderthecaseE[0;1].ThemonotonicityoftheGreenfunctionyieldsg(z)g

PAGE 8

(c.f.[1,Theorem3.6]).Varioussucientconditionsfor(1.0.1)intermsofmetricpropertiesofEarestatedin[4],wherethereadercanalsondfurtherreferences. TherearecompactsetsE[0;1]oflinearLebesguemeasure0withproperty(1.0.1)(seee.g.[4,Corollary5.2]),hence(1.0.1)mayhold,thoughthesetEisnotdenseat0intermsoflinearmeasure.Onthecontrary,V.Andrievskii[2]provedthatifEsatises(1.0.1)thenitsdensityinasmallneighborhoodof0,measuredintermsoflogarithmiccapacity,isarbitraryclosetothedensityof[0;1]inthatneighborhood,i.e.(1.0.1)implieslimr!0cap(E\[0;r]) 4:(1.0.2) InChapter2wewillproveanintegralestimateforthedensityviatheGreenfunction,fromwhich(1.0.2)easilyfollows. AndrievskiialsoconstructedaregularcompactsetE[0;1]suchthatlimr!0g()]TJ/F4 11.95 Tf 9.3 0 TD[(r) 2 holdsbutliminfr!0cap(E\[0;r]) Furthermoreheprovedthatconversely,(1.0.2)doesnotimply(1.0.1). Nowlet'sturntothecaseE[)]TJ/F1 11.95 Tf 9.29 0 TD[(1;1].Inthiscaseg(ir)g

PAGE 9

becauseg(x+iy)ismonotoneiny.Aswewillsee,thenecessaryconditionfortheoptimalsmoothnesscanbegeneralizedtothiscase,aswell. LetusconsidernowthemoregeneralsettingwhenEisanarbitrarycompactsubsetofC.Assumethat0isaboundarypointof.Severalequivalentconditionsareknownfortheregularityof0(seee.g.([13,AppendixA2.]).OneofthemisduetoWiener.ItcharacterizestheregularitywiththecapacityofthesetsEn=E\( withsomepositivenumbersC;. For">0setNE(")=fn2N:cap(En)"2)]TJ/F6 7.97 Tf 6.58 0 TD[(ng;(1.0.7) andwesaythatasubsequenceN=fn1
PAGE 10

Totiksuggestedthattheseresultscouldbeextendedtothehigherdimensionalcase,i.e.whenERd.ForthiscaseaWienertypeconditionlikeinTheorem1.0.2wasalreadydenedbyMaz'ja(see[7]-[10]).Maz'japroveditssuciencyfortheHoldercontinuityofthesolutiontotheDirichletproblemandshowedthatingeneralitisnotnecessary.InChapter3wewillprovethesuciencyofthisconditionfortheHoldercontinuityoftheGreenfunctionandshowthatitisalsonecessaryprovidedEsatisestheconecondition.Wealsogiveanequivalentcharacterizationintermsoftheequilibriummeasure.Inotherwords,undertheconeconditionwecompletelycharacterizeHoldercontinuity,whichhasbeenalongstandingopenproblem.4

PAGE 11

Forthenotionsoflogarithmicpotentialtheoryseee.g.[12]or[13].InwhatfollowsEdenotestheequilibriummeasureofE,U(z):=Zlog1 thelogarithmicpotentialofthemeasure,gG(z;a)theGreenfunctionofthedomainGwithpoleata,!(x;H;G)theharmonicmeasureinGcorrespondingtothesetH@G.WeshallfrequentlyusetherelationgCnE(z)=log1 cap(E))]TJ/F4 11.95 Tf 11.95 0 TD[(UE(z);z2CnE(2.1.1)5

PAGE 12

LetGbeadomainwithcompactboundaryandwithcap(@G)>0,andletbeameasuresupportedon forz2@Gwiththeexceptionofasetofcapacity0.ForregularGtheexceptionalsetisempty.IfGisbounded,thentheconstantis0([13,Ch.II,Theorem4.1]),andifGisunbounded,thenitis([13,Ch.II,Theorem4.4])const=ZGgG(a;1)d(a):(2.1.3) WeshallusethenotationBal;Gforthebalayagemeasure Thereisaconnectionbetweenharmonicandbalayagemeasures:ifK@Garecompactsets,thenforx2GtheequalityBalx;G(K)=!(x;K;G)(2.1.4) holds,wherexdenotesthepointmass(Diracmeasure)placedatthepointx(seee.g.[13,AppendixA3,(3.3)]).Therefore,inwhatfollowsweshallinterchangeablyusetheharmonicmeasureandbalayagenotations. Recallthatcap(I)=jIj=4foranyintervalI,wherejIjdenotesthelength(Lebesguemeasure)ofI.6

PAGE 13

4)]TJ/F1 11.95 Tf 13.15 8.09 TD[(cap(E"(t)) 4)]TJ/F1 11.95 Tf 13.15 8.09 TD[(cap(E"(t)) Sincecap(E")jE"j=4,condition(2.2.7)issomewhatweakerthanZ101 4)]TJ 13.15 8.08 TD[(jE"(t)j whichisknowntobesucientfor(1.0.1)(see[4,Theorem2.1]). Andrievskii'stheoremisaconsequenceofCorollary2.2.2(seeLemma2.5.1). Condition(2.2.7)isnotsucientforHoldercontinuity,itdoesnotimply(1.0.1).Indeed,letPk2k=1butPk3k<1,andconsiderasetEoftheformE=[0;1]n1[k=1((1)]TJ/F4 11.95 Tf 11.95 0 TD[(k)2)]TJ/F6 7.97 Tf 6.58 0 TD[(nk;2)]TJ/F6 7.97 Tf 6.59 0 TD[(nk) withsomeveryfastincreasingsequencefnkg(saynk+1>k2nk).Onecanverifythatforthisset(2.2.8)istruebecauseofPk3k<1,butitwasshownin([4,Corollary3.3])that(1.0.1)doesnothold,duetoP2k=1. ThemethodusedintheproofofTheorem2.2.1canbeappliedtothecaseE[)]TJ/F1 11.95 Tf 9.29 0 TD[(1;1]aswell.ThehighestsmoothnessoftheGreenfunctionattheorigin(Lipschitz7

PAGE 14

4)]TJ/F1 11.95 Tf 13.15 8.09 TD[(cap(E"(t)) 2:(2.2.11)

PAGE 15

4)]TJ/F1 11.95 Tf 13.14 8.08 TD[(cap(Fj) Fortheproofrstofallnoticethat(1.0.1)impliesE[[0;r]([0;r])C2g()]TJ/F4 11.95 Tf 9.3 0 TD[(r);00(recallthatE[[0;r]denotestheequilibriummeasureofE[[0;r]).Thisisimmediate,since(see(2.1.1))g()]TJ/F4 11.95 Tf 9.29 0 TD[(r)g cap(E[[0;r]))]TJ/F4 11.95 Tf 11.95 0 TD[(UE[[0;r]()]TJ/F4 11.95 Tf 9.29 0 TD[(r)=UE[[0;r](0))]TJ/F4 11.95 Tf 11.96 0 TD[(UE[[0;r]()]TJ/F4 11.95 Tf 9.3 0 TD[(r)=Zlogt+r tdE[[0;r](t)log2Zr0dE[[0;r](t)=(log2)E[[0;r]([0;r]): 4)]TJ/F1 11.95 Tf 13.15 8.08 TD[(cap(Fj)

PAGE 16

andtherelation(2.3.13)follows.

PAGE 17

4C1p 4C1p 4C1p Nowsince[0;r]isoutsideDj,Bal[0;1]

PAGE 18

Fory62E[[0;r]thequantityBaly;Cn(E[[0;r])([0;r])=!y;[0;r];Cn(E[[0;r]) withsomeabsoluteconstantc1>0becausedist(Jj;0)dist(Jj;[0;1])jIjj=dj.By([13,Ch.II,(4.47)])wehaveBal)]TJ/F6 7.97 Tf 6.58 0 TD[(dj;Cn(E[[0;r])([0;r])Bal)]TJ/F6 7.97 Tf 6.58 0 TD[(dj;Cn[0;1]([0;r])(2.3.19)=1

PAGE 19

"1 4)]TJ/F1 11.95 Tf 13.14 8.08 TD[(cap(E"(qm)) "1 4)]TJ/F1 11.95 Tf 13.15 8.09 TD[(cap(E"(qjM+l)) ButE"(t)=[0;"u=2][(E"(u)\["u=2;u])[[u;t],i.e.E"(t)isobtainedfromE"(t)\["u=2;u]byattachingone-oneintervalstotherightandtotheleft.Therefore,wecanapplyLemma2.5.4below(2.5.42),twicetoconcludecap(E"(u)\(["u=2;u])) 4)]TJ/F1 11.95 Tf 13.14 8.09 TD[(cap(E"(t)) 4)]TJ/F1 11.95 Tf 13.14 8.09 TD[(cap(E"(u)) Thisistrueforallutu(1)]TJ/F4 11.95 Tf 12.15 0 TD[("=2)=(1)]TJ/F4 11.95 Tf 12.14 0 TD[("),thereforeifwesquarebothsides,13

PAGE 20

4)]TJ/F1 11.95 Tf 13.15 8.08 TD[(cap(E"(t)) 1)]TJ/F4 11.95 Tf 11.95 0 TD[("1 4)]TJ/F1 11.95 Tf 13.14 8.08 TD[(cap(E"(u)) Letkbethelargestintegerforwhichqk>2r ".Summingup(2.3.24)forthevaluesu=q;q2;q3;:::;qkandmakinguseof(2.3.21)weobtainZ1qk1 4)]TJ/F1 11.95 Tf 13.14 8.09 TD[(cap(E"(t)) "1 "1)]TJ/F6 7.97 Tf 13.3 4.7 TD[(" "; ".Then,changing4r "forrwecanuseHarnack'sinequalitytoobtaing()]TJ/F4 11.95 Tf 10.49 8.09 TD[("r WearegoingtousethenotationsofStepI.Insteadof(2.3.14)nowwehaveE[[0;r]([0;r])C2g(ir);0
PAGE 21

tdE[[0;r](t)logp holdswithsomepositiveconstantc. Now(c.f.(2.3.16))wehaveBal[)]TJ/F8 7.97 Tf 6.58 0 TD[(1;1] sinced[)]TJ/F8 7.97 Tf 6.58 0 TD[(1;1](t)=1 In(2.3.19)weused)]TJ/F6 7.97 Tf 6.59 0 TD[(dj.Now,since)]TJ/F4 11.95 Tf 9.3 0 TD[(djmaybeinE,letuschangeitforidj.ByHarnack'sinequalitywehavefory2JjBaly;Cn(E[[0;r])([0;r])c1Balidj;Cn(E[[0;r])([0;r])c1Balidj;Cn[)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1]([0;r])=c1!idj;[0;r];Cn[)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1]:(2.4.28) Applyingthetransformation'(z)=z)]TJ 12.55 9.83 TD[(p

PAGE 22

2 22)]TJ/F8 7.97 Tf 6.59 0 TD[(arccosrZ3 1)]TJ/F1 11.95 Tf 11.95 0 TD[(2(q dj(q 1)]TJ/F1 11.95 Tf 11.96 0 TD[(2(q 1)]TJ/F1 11.95 Tf 11.95 -0.01 TD[((q 1)]TJ/F1 11.95 Tf 11.96 0 TD[((p

PAGE 23

21 4r dj: 1222jr; TheproofofCorollary2.2.5isimmediatefromLemmas2.5.1and2.5.4.Firstofall,Lemma2.5.1implies(1.0.2)andlimr!0cap(E\[)]TJ/F4 11.95 Tf 9.3 0 TD[(r;0]) 4: 2:(2.4.29) Next,takingI=[0;r],J=[)]TJ/F4 11.95 Tf 9.3 0 TD[(r;0],F=E\[0;r]andG=E\[)]TJ/F4 11.95 Tf 9.29 0 TD[(r;0]in(2.5.41)wecaninfercap(E\[)]TJ/F4 11.95 Tf 9.29 0 TD[(r;r]) cap(E\[)]TJ/F4 11.95 Tf 9.3 0 TD[(r;0])[[0;r]4cap(E\[0;r]) Finally,(2.2.11)isadirectconsequenceof(2.4.29)and(2.4.30). 17

PAGE 24

4)]TJ/F1 11.95 Tf 13.15 8.51 TD[(cap(E"=2(u)) 4)]TJ/F1 11.95 Tf 13.15 8.08 TD[(cap(E"(t)) 4)]TJ/F1 11.95 Tf 13.15 8.08 TD[(cap(E"(t)) 4)]TJ/F1 11.95 Tf 13.15 8.09 TD[(cap(E"(t)) 4)]TJ/F1 11.95 Tf 13.15 8.09 TD[(cap(E"(t)) log(1+)Ztt=(1+)1 4)]TJ/F1 11.95 Tf 13.14 8.52 TD[(cap(E"=2(u)) 4)]TJ/F1 11.95 Tf 13.15 8.09 TD[(cap(E"(t)) 4:(2.5.31) Nowletftngbeanarbitrarypositivesequencetendingto0andsetFn=E"(tn)=tn;n=1

PAGE 25

cap(Fn)(1+")log4;x2Fn;holds,whichimpliesfornmax(n0;n1)Un(x)1 1)]TJ/F1 11.95 Tf 11.95 0 TD[(2p cap(Fn\([";1)]TJ/F4 11.95 Tf 11.95 0 TD[("])ZUndn1 1)]TJ/F1 11.95 Tf 11.95 0 TD[(2p 4(1+")=(1)]TJ/F8 7.97 Tf 6.59 0 TD[(2p Intheprecedingargumentweusedthatasn!1,wehaveFn![0;1]intheweaktopologyonmeasures.Infact,letbeaweaklimitofsomesubsequence,sayFnl!asl!1.Thenissupportedin[0;1],hastotalmass1,andallwehavetoshowisthat=[0;1].WeknowthatUFn(x)=log1 cap(Fn)(2.5.32) forx2Fnwiththeexceptionofasetofcapacity0,andthesameistruefor[0;1].SinceFn[0;1],itfollowsthatUFn(x)U[0;1](x)+logcap([0;1]) cap(Fn) forx2Fnwiththeexceptionofasetofcapacity0,andsinceeverysetofzerocapacityhaszeroFn-measure(see[13,RemarkI.1.7,p.28]),itfollowsthatthisinequalityis19

PAGE 26

However,thefunctionU[0;1](x))]TJ/F4 11.95 Tf 11.95 0 TD[(U(x) vanishesatinnity,soitisharmonicthere,andanappealtotheminimumprincipleonthedomain 4)]TJ/F1 11.95 Tf 13.15 8.08 TD[(cap(F) 2 Proof.Itisenoughtoshowthatif1=8andH=fx2[)]TJ/F1 11.95 Tf 9.29 0 TD[(1;1]:!(x;T;DnF)>2=3g;

PAGE 27

Letthusbe1=8.Supposetothecontrarythat(2.5.33)isnottrue,i.e.jHj<4.Thencap([)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1]nH)j[)]TJ/F1 11.95 Tf 9.29 0 TD[(1;1]nHj LetD2denotetheopendiskabouttheoriginandofradius2.Ifx2[)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1]nH,wehave!(x;@D2;D2nF)!(x;T;DnF)<2 cap(F)+log3; cap(F))]TJ/F4 11.95 Tf 11.95 0 TD[(UF(x)2=log1 cap([)]TJ/F1 11.95 Tf 9.29 0 TD[(1;1]nH))]TJ/F4 11.95 Tf 11.95 0 TD[(U[)]TJ/F13 5.98 Tf 5.76 0 TD[(1;1]nH(x)+2: cap(F)log1 cap([)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1]nH)+2;21

PAGE 28

2)]TJ/F1 11.95 Tf 11.95 0 TD[(cap(F)1 2)]TJ/F1 11.95 Tf 11.95 0 TD[(cap([)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1]nH)e)]TJ/F8 7.97 Tf 6.59 0 TD[(2<1 2)]TJ/F1 11.95 Tf 13.15 8.09 TD[(1)]TJ/F1 11.95 Tf 11.95 0 TD[(2 2(1)]TJ/F4 11.95 Tf 11.96 0 TD[(e)]TJ/F8 7.97 Tf 6.58 0 TD[(2)+2

PAGE 29

2ZJ001)]TJ/F4 11.95 Tf 11.95 0 TD[(y2 2ZA001)]TJ/F4 11.95 Tf 11.96 0 TD[(y2 whichisclearwithsomec">0,sincey2[)]TJ/F1 11.95 Tf 9.29 0 TD[(1;1]andonthetwosidesduringintegrationrunstroughtwoarcsofcomparablelengthbothofwhichlieofdistanced"=4from[)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1].Thus,(2.5.36)istruewithsomec">0,andthisgives(2.5.35). Nextweturntothegeneralcase,i.e.when[)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1]nF=[)]TJ/F1 11.95 Tf 9.3 0 TD[(1+";1)]TJ/F4 11.95 Tf 12.92 0 TD[("]nFisanarbitraryopenset.Sincetheconstantc"shouldbeindependentofthesetF(dependingonlyon"with[)]TJ/F1 11.95 Tf 9.29 0 TD[(1;)]TJ/F1 11.95 Tf 9.29 0 TD[(1+"][[1)]TJ/F4 11.95 Tf 11.26 0 TD[(";1]F),withoutlossofgeneralitywemayassumeFtoconsistofnitelymanyintervals,inwhichcase[)]TJ/F1 11.95 Tf 9.29 0 TD[(1;1]nFconsistsofnitelymanyopenintervals,sayI1;:::;Im.23

PAGE 30

Weshowthattheconstantc"veriedaboveforthespecialcasewhen[)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1]nFwasaninterval,isappropriate.Tothisend,startingfrom0=x,wesuccessivelydenethemeasuresnbyn+1=Baln;Dn([)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1]nIjn);

PAGE 31

NowBalx;DnFistheuniquemeasuresupportedonT[Fwhichhasmass1anditslogarithmicpotentialislog1=jz)]TJ/F4 11.95 Tf 12.29 0 TD[(xj,thustheproofwillbecompleteifweshowthathasmass1,i.e.(T[F)=1,whichisthesameaslimn!1n([)]TJ/F1 11.95 Tf 9.3 0 TD[(1;1]nF)=0 whichwewantedtoproveanyway.Thiswillbedonebyshowingthatineachstep25

PAGE 32

LetIj=[aj;bj],andlet>0besosmallthatalltheintervals[aj)]TJ/F4 11.95 Tf 12.41 0 TD[(;aj]and[bj;bj+]arepartof()]TJ/F1 11.95 Tf 9.29 0 TD[(1;1)andtheyaredisjoint.ForI=Ijnandy2IthevalueBaly;Dn([)]TJ/F1 11.95 Tf 9.29 0 TD[(1;1]nI)([ajn)]TJ/F4 11.95 Tf 11.95 0 TD[(;ajn][[bjn;bjn+]); withsomepositiveconstantfollowsfromthefactthatherethelefthandsideis!(z;A)]TJ/F2 11.95 Tf 9.74 1.79 TD[([A+;D)=1 2ZA)]TJ/F9 7.97 Tf 6.25 1.07 TD[([A+1)]TJ/F4 11.95 Tf 11.95 0 TD[(z2 Weobtainfrom(2.5.40)Baly;Dn([)]TJ/F1 11.95 Tf 9.29 0 TD[(1;1]nI)(F)Baly;Dn([)]TJ/F1 11.95 Tf 9.29 0 TD[(1;1]nI)([ajn)]TJ/F4 11.95 Tf 11.95 0 TD[(;ajn][[bjn;bjn+]);26

PAGE 33

4cap(I[G):(2.5.41)

PAGE 34

cap(F))]TJ/F1 11.95 Tf 11.96 0 TD[(log4 TheanalogousformulaforF[GandI[Greadsaslog1 cap(F[G))]TJ/F1 11.95 Tf 11.95 0 TD[(log1 cap(I[G)=Z(I[G)n(F[G)g whereweusedthat(I[G)n(F[G)=InF,sotheintegrationisoverthesamesetontherighthandsidesof(2.5.43)and(2.5.44).SincethemeasureIisthebalayageofI[GontoI(see[13,TheoremIV.1.6,(e)]),wehaveonInFtheinequalitydI[G(a)dI(a).Atthesametimeg cap(F[G))]TJ/F1 11.95 Tf 11.95 0 TD[(log1 cap(I[G)log1 cap(F))]TJ/F1 11.95 Tf 11.95 0 TD[(log4 28

PAGE 35

ForthenotionsofclassicalpotentialtheoryinRdseee.g.[6].TheNewtonianpotentialofthemeasureisdenedasU(x):=Z1 infI(); whereSristhe(d)]TJ/F1 11.95 Tf 11.95 0 TD[(1)-dimensionalnormalizedsurfaceareameasureonSr.29

PAGE 36

whereq.e.means\quasi-everywhere",i.e.withtheexceptionofasetofzerocapacity. IfEisofpositivecapacity,thenEhasniteenergy.HenceasetofzerocapacityhaszeroE-measure,andsoifapropertyholdsquasi-everywhere,i.e.withtheexceptionofasetofzerocapacity,thenitalsoholdsE-almosteverywhere. IfisameasuresupportedonthecompactsetFandU(x)1forallx2Rd,thenthesetK:=fx:U(x)g(3.1.3) hascapacityatmost(1=)cap(F).Infact,ifKisofpositivecapacity,thentheinequalityU(x) cap(F)UK(x) cap(K)+ cap(K) holdstrueforquasi-everyx2K.HencethisistrueforK-almostallx,andthentheprincipleofdomination([6,Theorem1.27])givesthesameinequalityforallx2Rd.Nowcap(K)1 followsifweletxtendtoinnity. Weshallalsoneedthefollowingresult.ThereisapositiveconstantcsuchthatifAS1and(A)denotesthe(d)]TJ/F1 11.95 Tf 11.96 0 TD[(1)-dimensionalsurfaceareameasureofAthen(A)cp Indeed,ifdenotesthenormalizedsurfaceareameasureonS1thenbasedonthedenitionofcapacity:1 cap(A)1

PAGE 37

ZS1ZS11 k ForregularGtheexceptionalsetisempty.IfGisbounded,then Gandleavetherestofunchanged.InthissenseifG1G2,thentakingbalayageoutofG2canbedoneintwosteps:rsttakebalayageoutofG1,andthentakethebalayageoftheresultingmeasureoutofG2. Perhapsthemostimportantconnectionbetweenequilibriumandbalayagemea-suresisthefactthatifEFarecompactsetsofpositivecapacity,thenEisthebalayageofFontoE(i.e.outoftheunboundedcomponentof IfK@Garecompactsetsofpositivecapacity,thentheharmonicmeasure!(x;K;G)istheuniquesolutionofthegeneralizedDirichlet-probleminGcorre-spondingtothecharacteristicfunctionofKin@G.Thereisaconnectionbetween31

PAGE 38

holds,whereadenotesthepointmass(Diracmeasure)placedatthepointaand Green'sfunctionofGwithpoleaty2GisdenedasgG(x;y)=Uy(x))]TJ/F4 11.95 Tf 11.95 0 TD[(U x(y) wherey2SrandSristhenormalizedsurfaceareameasureonSr.Indeed,Poisson'sformula(seee.g.[3,Section1.3,(1.3.1)])givesd x(y) x(y) MultiplyingbyRd)]TJ/F8 7.97 Tf 6.59 0 TD[(2andlettingR!1wegetthat

PAGE 39

foranyalyingintheunboundedcomponentofRdnEwithsomeconstantsca,Ca. LetbeameasureonSr.ThelowerRadon-Nikodymderivative(density)ofwithrespecttonormalizedsurfaceareameasureonSrisdenedasfollows(seee.g.[5,Chapter3]or[11,ChapterVII]).Letx02Srand0<<1.ThentheconeC(x0;):=fx2Rd:hx;x0i Finally,letusrecallthattheNewtoniancapacityissubadditive:ifF=[ki=1Fi,thencap(F)kXi=1cap(Fi):(3.1.11) Inparticular,oneofthesetsFimusthavecapacitycap(F)=k.Ontheotherhand,ifthedistancebetweenthesetsF1andF2isatleastl,thencap(F1[F2)cap(F1)+cap(F2) 1+2cap(F1)cap(F2) Indeed,set=1)]TJ/F4 11.95 Tf 11.95 0 TD[(t

PAGE 40

LetBr=Br(0)betheballofradiusrabouttheorigin,andweshalldenoteitsclosureby

PAGE 41

withsomepositivenumbersC;. Followingthedenitionsin[4],for">0setNE(")=fn2N:cap(En)"2)]TJ/F6 7.97 Tf 6.58 0 TD[(n(d)]TJ/F8 7.97 Tf 6.58 0 TD[(2)g;(3.2.15) andwesaythatasubsequenceN=fn10,whichmeansthatcontainsaconewithvertexat0.Theorem3.2.1a)

PAGE 42

If,inaddition,Gsatisestheconeconditionat0,then1)-2)arealsoequivalentto3) LetFbeacompactsetsuchthat0isontheboundaryoftheunboundedcomponentofRdnF,andletbdenotethebalayageofsomemeasureoutofRdn(F[ 8d)]TJ/F8 7.97 Tf 6.58 0 TD[(21)]TJ/F4 11.95 Tf 15.76 8.09 TD[("

PAGE 43

B1+ F; Thelefthandsideiswhatisontheleftof(3.3.18),andsinceS8=S8=8d)]TJ/F8 7.97 Tf 6.59 0 TD[(2,and Foreverya2S8(3.1.9)withr=1andR=8showsthat 9d)]TJ/F8 7.97 Tf 6.58 0 TD[(1S1>1 9dS1; 9dS1;(3.3.20) andwehavetoestimatehowmuchofcS8goesontoF.SinceweassumedF2=F[ 8d)]TJ/F8 7.97 Tf 6.58 0 TD[(2F2(F):(3.3.21) ThedistanceofthesetsF 1+2cap(F)cap( 3d)]TJ/F13 5.98 Tf 5.75 0 TD[(2(cap(F)+cap( 1+2 3d)]TJ/F13 5.98 Tf 5.75 0 TD[(2cap(F);

PAGE 44

3d)]TJ/F13 5.98 Tf 5.76 0 TD[(24"1+": 8d)]TJ/F8 7.97 Tf 6.58 0 TD[(2" 8d)]TJ/F8 7.97 Tf 6.59 0 TD[(2S1)]TJ/F1 11.95 Tf 20.98 8.09 TD[(1 8d)]TJ/F8 7.97 Tf 6.59 0 TD[(2" 8d)]TJ/F8 7.97 Tf 6.59 0 TD[(21)]TJ/F4 11.95 Tf 15.76 8.09 TD[(" Weshalluse(3.3.18)inascaledform,namelyifEiscompact,0isontheboundaryoftheunboundedcomponentofRdnEandcap(E\( thenwehave[S2)]TJ/F12 5.98 Tf 5.76 0 TD[(n 8d)]TJ/F8 7.97 Tf 6.58 0 TD[(21)]TJ/F4 11.95 Tf 15.76 8.08 TD[(" wherenowbdenotesbalayageoutofRdn(E[ AfterthispreparationletusreturntothesetNE(")whichwasassumedtobeofpositivelowerdensity.Thenthereisan>0suchthatforlargeNthesetNE(")hasatleastNelementssmallerthanN.ForlargeNthenwecanselectasubsetKNE(")\f2;:::;N)]TJ/F1 11.95 Tf 11.95 0 TD[(2g38

PAGE 45

WesetEn=E[ Firstnotethat0 S2)]TJ/F12 5.98 Tf 5.75 0 TD[(n=c1 2d)]TJ/F8 7.97 Tf 6.58 0 TD[(2S2)]TJ/F12 5.98 Tf 5.75 0 TD[(n)]TJ/F13 5.98 Tf 5.76 0 TD[(1:(3.3.24) Ontheotherhand,ifn=ni)]TJ/F1 11.95 Tf 10.82 0 TD[(1,withni2K,then(3.3.22)istrue,henceforsuchnwehave(3.3.23).Again,themeasuren+3isthebalayageofnontoEn+3,andintakingthisbalayagewesweepoutonlythepartofnthatissittingonS2)]TJ/F12 5.98 Tf 5.75 0 TD[(nnE.Thus,ifbdenotesthebalayageontoEn+3,thenn+3 8d)]TJ/F8 7.97 Tf 6.58 0 TD[(21)]TJ/F4 11.95 Tf 15.76 8.08 TD[("

PAGE 46

8d)]TJ/F8 7.97 Tf 6.58 0 TD[(21)]TJ/F4 11.95 Tf 15.76 8.08 TD[(" 2d)]TJ/F8 7.97 Tf 6.58 0 TD[(2N)]TJ/F8 7.97 Tf 6.59 0 TD[(3kS2)]TJ/F12 5.98 Tf 5.76 0 TD[(N(S2)]TJ/F12 5.98 Tf 5.75 0 TD[(N)1 2N(d)]TJ/F8 7.97 Tf 6.59 0 TD[(2)1)]TJ/F4 11.95 Tf 15.76 8.08 TD[(" 2N(d)]TJ/F8 7.97 Tf 6.59 0 TD[(2)1)]TJ/F4 11.95 Tf 15.76 8.08 TD[(" 2N(d)]TJ/F8 7.97 Tf 6.58 0 TD[(2)1)]TJ/F4 11.95 Tf 15.77 8.08 TD[(" andisdenedbytheequation2)]TJ/F6 7.97 Tf 6.58 0 TD[(=1)]TJ/F4 11.95 Tf 15.76 8.08 TD[("

PAGE 47

Fornotationalconveniencelet1=B2)]TJ/F13 5.98 Tf 5.75 0 TD[(1,L=B2)]TJ/F12 5.98 Tf 5.75 0 TD[(L,L+1=B2)]TJ/F12 5.98 Tf 5.76 0 TD[(L)]TJ/F13 5.98 Tf 5.76 0 TD[(1,3=2=B32)]TJ/F12 5.98 Tf 5.75 0 TD[(L)]TJ/F13 5.98 Tf 5.75 0 TD[(2,andlettheboundingsurfacesoftheseballsbedenotedbyT1,TL,TL+1andT3=2,respectively.SetalsoF3=2=F\ isthebalayageoutofRdn WestartfromtherepresentationF=L+1[j=2F\( Nowleta21n WithoutlossofgeneralitywemayassumethatFisofpositivecapacity(otherwise41

PAGE 48

F=ZU Weshallneedasimilarreasoningforthebalayage=a:=eaofaoutof(Rdn Thus,ifK:=a:UF(a)1 thenfora62Kwehave(3.3.29),andifK3=2:=a:UF3=2(a)1 thenfora2TLnK3=2wehave(3.3.30).ForthecapacityofKwegetfrom(3.1.3){42

PAGE 49

IfLdenotesthe(d)]TJ/F1 11.95 Tf 10.85 0 TD[(1)-dimensionalnormalizedsurfaceareameasureonTL,thenby(3.1.5)wehaveL(K\TL)cp AnidenticalinequalityistrueforK3=2(c.f.(3.3.34)):L(K3=2\TL)c2L=2p Leta;b2TL,andletea,ebbethebalayageofa;boutofthedomainRdn(F3=2[ 1)]TJ/F1 11.95 Tf 11.95 0 TD[(3=4db=7db;

PAGE 50

Chooseandxab2TLnK3=2(see(3.3.32)).By(3.3.36)if"issucientlysmallcomparedtoL,thenthereissuchab.Inthiscase(3.3.30)giveseb(F3=2)1=L,hencethebalayage:= thenL+1(H)1 foralla2TLandallx2TL+1nH. Nextconsiderthebalayagea:= c 2L+1d)]TJ/F8 7.97 Tf 6.58 0 TD[(23=2L+2+1=2L+1 c

PAGE 51

Inasimilarfashionweobtainfora:= (2L)]TJ/F1 11.95 Tf 11.95 0 TD[(1)d)]TJ/F8 7.97 Tf 6.58 0 TD[(1ba(T1)c0(2L+1) (2L)]TJ/F1 11.95 Tf 11.96 0 TD[(1)d)]TJ/F8 7.97 Tf 6.59 0 TD[(1:(3.3.41) Nownotethata+a+a+ a(y) (2L)]TJ/F1 11.95 Tf 11.95 0 TD[(1)d)]TJ/F8 7.97 Tf 6.59 0 TD[(1; 2d(L+1)ja)]TJ/F4 11.95 Tf 11.95 0 TD[(yjd)]TJ/F4 11.95 Tf 17.23 8.09 TD[(c1 Thisderivationusedtheexistenceofb2TLnK3=2,anditisvalidif"issucientlysmallcomparedtoL(see(3.3.36)).StepII Let>0.SupposethatisameasureonTLsuchthatd(y) whereHTLisof(normalizedsurfacearea)measureatmost.Letbbethebalayageofoutof(1n

PAGE 52

2d)]TJ/F8 7.97 Tf 6.59 0 TD[(2(1)]TJ/F4 11.95 Tf 11.95 0 TD[(c2)fory2TL+1nH;(3.3.44)whereHisasetof(normalizedsurfacearea)measureatmostandc2>0isaconstantdependingonlyond. Firstofallnotethatwehave(3.3.42)fora2TLnKandy2TL+1nH,whereHisthexedsetdenedin(3.3.38),andalsonotethattheintegraloverTL+1ofthersttermontherightof(3.3.42)withrespecttodLisZTL3 2d(L+1)ja)]TJ/F4 11.95 Tf 11.95 0 TD[(yjddL(a)=ZTLd a(y) L(y) TL(y) 2(L+1)(d)]TJ/F8 7.97 Tf 6.59 0 TD[(2)dTL+1=1 2d)]TJ/F8 7.97 Tf 6.58 0 TD[(2; 2d(L+1)ja)]TJ/F4 11.95 Tf 11.95 0 TD[(yjd)]TJ/F4 11.95 Tf 17.23 8.09 TD[(c1 2d)]TJ/F8 7.97 Tf 6.58 0 TD[(2)]TJ/F4 11.95 Tf 17.22 8.09 TD[(c1 Wewritewitha2TLandy2TL+1db(y) 2d)]TJ/F8 7.97 Tf 6.58 0 TD[(2)]TJ/F4 11.95 Tf 17.23 8.09 TD[(c1 2d(L+1)ja)]TJ/F4 11.95 Tf 11.95 0 TD[(yjddL(a):

PAGE 53

2d)]TJ/F8 7.97 Tf 6.58 0 TD[(2)]TJ/F4 11.95 Tf 17.23 8.09 TD[(c1 2d)]TJ/F8 7.97 Tf 6.58 0 TD[(2(1)]TJ/F1 11.95 Tf 11.96 0 TD[(2d);(3.3.46) providedc1 ThisconditionshouldbeunderstoodinthesensethatrstwechooseLlargeenough,thenforxedLchoose">0smalltosatisfy(3.3.47).Furthermore,assumingthiscondition,HhasmeasureatmostL+1(H)1 Thus,withsuchachoiceforLand"theestimate(3.3.44)holdswithc2=2d.StepIII Letc3=4>>0,wherec3isaconstanttobechosenlater,andsupposethatisameasureonTLsuchthatd(y) whereH0isof(normalizedsurfacearea)measureatleastc3)]TJ/F4 11.95 Tf 12.19 0 TD[(.Thus,weconsider47

PAGE 54

Let,asbefore,bbethebalayageofoutof(1n whereHisasetof(normalizedsurfacearea)measureatmostandc4dependsonlyond. ForaproofjustfollowtheproofinstepII.Wehave(3.3.42)fora2TLnKwithHgivenin(3.3.38),andnotethat3 2d(L+1)ja)]TJ/F4 11.95 Tf 11.95 0 TD[(yjd3 22d: 22d)]TJ/F4 11.95 Tf 17.23 8.08 TD[(c1 NotethatbothoftheseconditionsaresatisedifLissucientlylargeand"issucientlysmall. 48

PAGE 55

kxk1)]TJ/F4 11.95 Tf 11.96 0 TD[(g:(3.3.51) Considerthedomain(B2n Let0insuchawaythatalltheestimatesinstepsII{IVhold. LetEn=E[

PAGE 56

WewanttocompareAn+1withAnfornL.Let Inacompletelysimilarmanner,ifn21,thenAnisofthesecondtypewhileAn+1isofthersttype,i.e.vn(y)Anforally2S2)]TJ/F12 5.98 Tf 5.75 0 TD[(n\Cwiththeexceptionofasetofmeasureandvn+1(y)An+1forally2S2)]TJ/F12 5.98 Tf 5.76 0 TD[(n)]TJ/F13 5.98 Tf 5.75 0 TD[(1withtheexceptionofasetofmeasure.Nowinsteadof(3.3.44)weapply(3.3.50)toconcludethatAn+1c4An. Finally,ifn22,nL,thenAn+1isdenitelyofthesecondtype,butAn50

PAGE 57

Insummary,wehaveAn+1(1=2d)]TJ/F8 7.97 Tf 6.58 0 TD[(2)(1)]TJ/F4 11.95 Tf 12.44 0 TD[(c2)Anforn20,An+1c4Anforn21andAn+1(c3c=2)Anforn22.Ifs=sNdenotesthenumberofelementsofNE(")[f0gnotlargerthanN,thenthereareatmostselementsof1andatmostsLelementsof2notlargerthanN.Thus,wecanconcludeAN+11 2d)]TJ/F6 7.97 Tf 6.59 0 TD[(sN(1)]TJ/F4 11.95 Tf 11.95 0 TD[(c2)N(c4)sNc3c ForallsuchNwecanconcludethatAN+1(2=c3)(1=2d)]TJ/F8 7.97 Tf 6.58 0 TD[(2)N(1)]TJ/F4 11.95 Tf 11.95 0 TD[(c2)2N; 2d)]TJ/F8 7.97 Tf 6.59 0 TD[(2N(1)]TJ/F4 11.95 Tf 11.95 0 TD[(c2)2N(3.3.54) because,independentlyifANisoftherstorsecondtype,wehavevN(y)ANonasetofmeasureatleastc3)]TJ/F4 11.95 Tf 11.95 0 TD[(c3=2. Nowwecaneasilycompletetheproof.LetN+1betheunboundedcomponentofRdnEN+1.ConsiderGreen'sfunctionwithpoleaty02N+1andintegrateitoverthesphereSrwithr=rN=2)]TJ/F6 7.97 Tf 6.58 0 TD[(N.ZSrgN+1(x;y0)dSr(x)=ZSr)]TJ/F4 11.95 Tf 5.48 -9.68 TD[(gN+1(x;y0))]TJ/F4 11.95 Tf 11.95 0 TD[(gN+1(0;y0)dSr(x)=ZSr)]TJ/F4 11.95 Tf 5.48 -9.68 TD[(Uy0(x))]TJ/F4 11.95 Tf 11.95 0 TD[(Uy0(0)dSr(x)51

PAGE 58

whereedenotesthebalayageoutofN+1. Heretherstintegrandis1 wherec6dependsonlyony0andd.ForthesecondintegralwehaveZSrUfy0(0))]TJ/F4 11.95 Tf 11.95 0 TD[(Ufy0(x)dSr(x)=ZEN+11 SinceZSr1 provided<(1)]TJ 11.95 9.89 TD[(p

PAGE 59

log2log1 1)]TJ/F4 11.95 Tf 11.96 0 TD[(c2: TheproofaboveusedthatEiscontainedintheunitballandomitstheconeC2.Thegeneralcasecanbesimilarlyhandled.Infact,letcontaintheconeC=C(x0;2;`).SelectasphereSr0,r0
PAGE 60

Conversely,suppose2).Letjxj=rbesmall,andE=EnB2r.Let a(y)):

PAGE 61

Nextweshowthat3)implies1).LetRbesolargethat@GBRandconstructadomainTinthefollowingway.Ifa62 Itislefttoshow1))3undertheconecondition.Undertheconecondition1)impliesthepositivelowerdensityofNE(")forsome">0,i.e.thereisanandanN1suchthatjNE(")\f0;1;:::;Ngj4NforNN1.ThenforlargeN,sayforNN2,wealsohavejNE(")\f[(2)N]+1;[(2)N]+2;:::;MgjM

PAGE 62

56

PAGE 63

L.V.Ahlfors(1973):ConformalInvariants.NewYork:McGraw-Hill.[2] V.V.Andrievskii,ThehighestsmoothnessoftheGreen'sfunctionimpliesthehighestdensityoftheset(manuscript)[3] D.H.Armitage,S.J.Gardiner,ClassicalPotentialTheory,Springermonographsinmathematics,Springer-Verlag,London,2001.[4] L.Carleson,V.Totik,HoldercontinuityofGreen'sfunctions,ActaSci.Math.,Szeged,toappear[5] L.L.Helms,IntroductiontoPotentialTheory,Pureandappliedmathematics,XXII,Wiley-Interscience,NewYork,1969.[6] N.S.Landkof,FoundationsofModernPotentialTheory,Grundlehrendermath-ematischenWissenschaften,180,SpringerVerlag,Berlin,1972.[7] V.G.Maz'ja,Regularityattheboundaryofsolutionsofellipticequations,andcomformalmappings,Dokl.Akad.Nauk.SSSR,152,1297-1300(Russian)1963.[8] V.G.Maz'ja,OnthemodulusofcontinuityofthesolutionsoftheDirichletproblemnearirregularboundary,in:Problemsinmathematicalanalysis;Bound-aryproblemsandintegralequations,Izdat.LeningradUniv.,Leningrad,45-48.(Russian)1966.[9] V.G.Maz'ja,Ontheregularityofboundarypointsforellipticequations,in:Investigationsonlinearoperatorsandfunctiontheory;99unsolvedproblems57

PAGE 64

V.G.Maz'ja,Onthemodulusofcontinuityofaharmonicfunctionataboundarypoint,Zap.Nauchn.Sem.Leningrad.Otdel.Mat.Inst.Steklov.(LOMI),135,87-95.(Russian)1984.[11] M.E.Munroe,MeasureandIntegration,Addison-Wesleymathematicsseries,Addison-Wesley,Reading,1959.[12] T.Ransford:PotentialTheoryintheComplexplane.Cambridge:CambridgeUniversityPress,1995.[13] E.B.Sa,V.Totik,LogarithmicPotentialswithExternalFields,GrundlehrendermathematischenWissenschaften,316,Springer-Verlag,NewYork/Berlin,1997.58

PAGE 65

FerencreceivedhisM.S.degreeinMathematicsattheUniversityofSzeged,Hun-garyin2000.HeenteredthePh.D.programinMathematicsattheUniversityofSouthFloridain2001.HiseldisPotentialTheoryandApproximationTheory,hisadvisorwasDr.VilmosTotik.WhileinthePh.D.program,FerenctaughtseveralundergraduatecoursesasaTeachingAssistant. Ferenclikesallkindsofsports,especiallytennis,pingpongandsoccer.


printinsert_linkshareget_appmore_horiz

Download Options

close
Choose Size
Choose file type
Cite this item close

APA

Cras ut cursus ante, a fringilla nunc. Mauris lorem nunc, cursus sit amet enim ac, vehicula vestibulum mi. Mauris viverra nisl vel enim faucibus porta. Praesent sit amet ornare diam, non finibus nulla.

MLA

Cras efficitur magna et sapien varius, luctus ullamcorper dolor convallis. Orci varius natoque penatibus et magnis dis parturient montes, nascetur ridiculus mus. Fusce sit amet justo ut erat laoreet congue sed a ante.

CHICAGO

Phasellus ornare in augue eu imperdiet. Donec malesuada sapien ante, at vehicula orci tempor molestie. Proin vitae urna elit. Pellentesque vitae nisi et diam euismod malesuada aliquet non erat.

WIKIPEDIA

Nunc fringilla dolor ut dictum placerat. Proin ac neque rutrum, consectetur ligula id, laoreet ligula. Nulla lorem massa, consectetur vitae consequat in, lobortis at dolor. Nunc sed leo odio.