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Lester L. Helms
Potential Theory
123
LesterL.Helms
UniversityofIllinois
DepartmentofMathematics
1409W.GreenSt.
UrbanaIL61801,USA
[email protected]
Editorialboard:
SheldonAxler,SanFranciscoStateUniversity
VincenzoCapasso,UniversitàdegliStudidiMilano
CarlesCasacuberta,UniversitatdeBarcelona
AngusMacIntyre,QueenMary,UniversityofLondon
KennethRibert,UniversityofCalifornia,Berkeley
ClaudeSabbah,CNRS,ÉcolePolytechnique
EndreSüli,UniversityofOxford
WojborWoyczyn´ski,CaseWesternReserveUniversity
ISBN978-1-84882-318-1 e-ISBN978-1-84882-319-8
DOI10.1007/978-1-84882-319-8
SpringerDordrechtHeidelbergLondonNewYork
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Dedicated to my wife Dorothea Helms
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Preface
The first six chapters of this book are revised versions of the same chapters
in the author’s 1969 book, Introduction to Potential Theory. At the time of
thewritingofthatbook,Ihadaccesstoexcellentarticles,books,andlecture
notes by M. Brelot. The clarity of these works made the task of collating
them into a single body much easier. Unfortunately, there is not a similar
collection relevant to more recent developments in potential theory. A new-
comerto the subjectwill find the journalliteratureto be a maze ofexcellent
papers and papers that never should have been published as presented. In
the Opinion Column of the August, 2008, issue of the Notices of the Amer-
ican Mathematical Society, M. Nathanson of Lehman College (CUNY) and
(CUNY) Graduate Center said it best “...When I read a journal article, I
often find mistakes. Whether I can fix them is irrelevant. The literature is
unreliable.”Fromtime totime,someonemusttrytofind apaththroughthe
maze.
In planning this book, it became apparent that a deficiency in the 1969
book would have to be corrected to include a discussion of the Neumann
problem, not only in preparation for a discussion of the oblique derivative
boundary value problem but also to improve the basic part of the subject
matter for the end users, engineers, physicists, etc. Generally speaking, the
choice of topics was intended to make the book more pragmatic and less
esoteric while exposing the reader to the major accomplishments by some
of the most prominent mathematicians of the eighteenth through twentieth
centuries. Most of these accomplishments had their origin in practical mat-
ters as, for example, Green’s assumption that there is a function, which he
called a potential function, which could be used to solve problems related to
electromagnetic fields.
This book is targetedprimarily at students with a backgroundin a senior
or graduate level course in real analysis which includes basic material on
topology, measure theory, and Banachspaces. I have tried to present a clear
path from the calculus to classic potential theory and then to recent work
vii
viii Preface
on elliptic partial differential equations using potential theory methods. The
goalhasbeento movethe readerinto a fertile areaofmathematicalresearch
as quickly as possible.
The author is indebted to L. Ho¨rmander and K. Miller for their prompt
responsestoqueriesaboutthedetailsofsomeoftheirproofs.Theauthoralso
thanksKarenBorthwickofSpringer,UK,forguidingtheauthorthroughthe
publication process.
Urbana, Illinois Lester L. Helms
September, 2008
Contents
0 Preliminaries ............................................. 1
0.1 Notation .............................................. 1
0.2 Useful Theorems ....................................... 4
1 Laplace’s Equation ....................................... 7
1.1 Introduction ........................................... 7
1.2 Green’s Theorem ....................................... 8
1.3 Fundamental Harmonic Function ......................... 9
1.4 The Mean Value Property ............................... 10
1.5 PoissonIntegral Formula ................................ 14
1.6 Gauss’ Averaging Principle .............................. 21
1.7 The Dirichlet Problem for a Ball ......................... 24
1.8 Kelvin Transformation .................................. 32
1.9 PoissonIntegral for Half-space ........................... 33
1.10 Neumann Problem for a Disk ............................ 39
1.11 Neumann Problem for the Ball........................... 42
1.12 Spherical Harmonics .................................... 49
2 The Dirichlet Problem.................................... 53
2.1 Introduction ........................................... 53
2.2 Sequences of Harmonic Functions......................... 54
2.3 Superharmonic Functions................................ 59
2.4 Properties of Superharmonic Functions.................... 65
2.5 Approximation of Superharmonic Functions................ 71
2.6 Perron-Wiener Method.................................. 75
2.7 The Radial Limit Theorem .............................. 88
2.8 Nontangential Boundary Limit Theorem................... 92
2.9 Harmonic Measure...................................... 100
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