Classical Potential Theory

343 Pages·2001·9.692 MB·English

by  David H. Armitage, Stephen J. Gardiner (auth.)

Springer Monographs in Mathematics
SSpprriinnggeerr--VVeerrllaagg LLoannddoann LLttdd..
David H. Armitage Stephen J. Gardiner
Classical Potential
Theory
, Springer
David Н. Armitage
Department of Pure Mathematics
Queen's University Belfast
Belfast
BT71NN
Northern Ireland
Stephen]. Gardiner
Department of Mathematics
University College Dublin
Dublin4
Ireland
ISBN 978-1·4471·1116·0
British Library Cataloguing in Publication Data
Armitage, D8vid Н.
Classical potential theory. - (Springer monographs in
mathematics)
1. Potential theory (Mathematics) 2. Нarmоniс functions
1. Title п. Gardiner. Stephen J.
515.9
IS8N 978-1-4471-1116-0
Library оССоngress Cataloging-in-Pub1ication Data
Апnitagе. David Н .• 1945-
C1assical potential theory. I David Н. Armitage. Stephen J. Gardiner.
р. cm. --(Springer тоnograрЬв in mathematics)
Indudes bibliographical references 8nd indaeI.
ISBN 978-1-4471-1116-0 ISBN 978-1-4471-0233-5 (eBook)
DOI 10.1007/978-1-4471-0233-5
1. Potential theory (Mathematics) 1. Gardiner. Stephen J. п. Title. III. SeПes.
QA404.7 .А65 2000
515.9-dc21
DQ..056312
Mathematics Subject Classilication (2000): 31.{)2; 31ВО5. 31В15. 31B2S. 31АOS. 3ОС85. 41А30
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To
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Preface
This book is about the potential theory of Laplace's equation,
82h 82h 82h
8 2 + 8 2 + ... + -82 = 0,
Xl X2 xN
in Euclidean space ~N, where N ~ 2; in brief, classical potential theory.
It involves the whole circle of ideas concerning harmonic and subharmonic
functions, maximum principles and analyticity, Green functions, potentials
and capacity, the Dirichlet problem and boundary integral representations.
From its origins in Newtonian physics, the subject has developed into a ma
jor field of research in its own right, intimately connected with several other
areas of real and complex analysis. Over the past half-century, new lines of
investigation have emerged and come to maturity, largely inspired by classi
cal potential theory: examples are non-linear potential theory, probabilistic
potential theory, axiomatic potential theory and pluripotential theory. For a
proper appreciation of these subjects an understanding of the classical the
ory is essential. There is also a close relationship between potential theory
in the plane and complex analysis: concepts from potential theory are im
portant and natural tools for the study of holomorphic functions. Further,
this connection suggests potential theoretic analogues of theorems concerning
functions of one complex variable, ranging from elementary results such as
the maximum modulus theorem and Laurent's theorem, to the approximation
theorems of Runge and Mergelyan and the theory of prime ends.
We treat our subject at a level intended to be accessible to graduate stu
dents. Prerequisite knowledge does not go beyond what is commonly taught
in undergraduate or first-year graduate courses. The reader will need a good
grasp of the limiting processes of analysis, some facility with calculus in
higher dimensions, and some measure theory. A few well-known theorems
from functional analysis are required, and only very basic topology and lin
ear algebra. Some of the less elementary results that are employed are stated
in the Appendix, where convenient references to proofs are supplied. As we
sometimes indicate connections with the theory of holomorphic functions,
familiarity with the rudiments of one-variable complex analysis would enrich
the reader's appreciation of this aspect of the subject.
We have set out to present rigorously and economically many of the re
sults and techniques that are central to potential theory and are the everyday
vii
VIll Preface
tools of researchers in the field. Occasionally we have taken the opportunity to
present some lesser known results that we have found useful and interesting.
The collection of theorems in Chapter 3 connecting convexity and subhar
monicity, some of which are not widely known but have elegant proofs, falls
into this category; another example from the same chapter is the characteri
zation of open sets in which the maximum principle holds (and, surprisingly,
these include some unbounded domains). In our own research we have some
times needed a standard result in a form not easily found in the literature.
This is no doubt a common experience, so we have given strong and general
versions of theorems when it has been feasible to do so without excessively
prolonging proofs. For example, the Dirichlet problem is discussed for the
most general open sets possible (which, when N ~ 3, include all open sets),
and the main removable singularity result (Theorem 5.2.1) does not require
that the exceptional polar set is closed. Obviously, we have had to decide to
omit certain topics, and among these are the notion of energy, and families
of capacities associated with various function spaces.
The first six chapters are of quite a concrete character, dealing with har
monic and subharmonic functions and potentials, and their particular prop
erties. Here the underlying topology is always the standard Euclidean one.
Each of these chapters concludes with a set of exercises, some fairly rou
tine and others leading step-by-step to results from the research literature.
The material in these chapters is especially appropriate to readers seeking a
background knowledge of the subject for wider application. In the final three
chapters the level of abstraction deepens as we introduce topological concepts
specially created for potential theory, such as the fine topology, the Martin
boundary and minimal thinness. Our aim here is to give the reader a firm
grounding in these more advanced topics on which to base future reading and
research. At the back of the book we have provided brief historical notes for
each chapter indicating, to the best of our knowledge, the original sources of
results and ideas, and pointing to further developments which lie beyond the
scope of this book.
In preparing this book we have, of course, benefitted from the work of
earlier authors. In particular, we acknowledge our indebtedness to Brelot [12,
1965], Helms [1, 1969], Hayman and Kennedy [1, 1976], Doob [6, 1984] and
Axler, Bourdon and Ramey [1, 1992]. Other related texts include Brelot [13,
1971]' Landkof [1,1972], Hayman [2,1989]' Ransford [1,1995]' and the older
works of Kellogg [1, 1929], Rad6 [1, 1937] and Tsuji [1, 1959]. We are also
grateful to Professors Hiroaki Aikawa, Ivan Netuka and Jifi Vesely for reading
various parts of the manuscript in draft form and making helpful suggestions.
Any defects that remain are, of course, the responsibility of the authors. Fi
nally, we express our appreciation to Michael Elliott, Sheila O'Brien, Siobhan
Purcell, Gerhard Schick and Thomas Unger for their careful typesetting of the
book, and to the staff of Springer-Verlag (UK) for their courteous efficiency
and helpfulness.
Contents
Notation and Terminology xiii
1. Harmonic Functions
1.1. Laplace's equation ............. ....................... 1
1.2. The mean value property ............. ................. 3
1.3. The Poisson integral for a ball .................... ...... 6
1.4. Harnack's inequalities ................................. 13
1.5. Families of harmonic functions: convergence properties 15
1.6. The Kelvin transform ................................. 19
1. 7. Harmonic functions on half-spaces ...................... 22
1.8. Real-analyticity of harmonic functions ................... 26
1. 9. Exercises ............................................ 30
2. Harmonic Polynomials
2.1. Spaces of homogeneous polynomials ..................... 33
2.2. Another inner product on a space of polynomials ......... 35
2.3. Axially symmetric harmonic polynomials ................ 37
2.4. Polynomial expansions of harmonic functions ............. 40
2.5. Laurent expansions of harmonic functions ................ 44
2.6. Harmonic approximation .............................. 47
2.7. Harmonic polynomials and classical polynomials .......... 53
2.8. Exercises ............................................ 55
3. Subharmonic Functions
3.1. Elementary properties ................................. 59
3.2. Criteria for subharmonicity ............................ 64
3.3. Approximation of subharmonic functions by smooth ones .. 68
3.4. Convexity and subharmonicity ......................... 72
3.5. Mean values and subharmonicity ...... ................. 75
3.6. Harmonic majorants .................................. 79
3.7. Families of subharmonic functions: convergence properties 82
3.8. Exercises ............................................ 84
IX
x Contents
4. Potentials
4.1. Green functions ...................................... 89
4.2. Potentials ............................. ............... 96
4.3. The distributional Laplacian ........................... 100
4.4. The Riesz decomposition .................... .......... 105
4.5. Continuity and smoothness properties ................... 109
4.6. Classical boundary limit theorems ....... ............... 112
4.7. Exercises ............................................ 118
5. Polar Sets and Capacity
5.1. Polar sets ............................................ 123
5.2. Removable singularity theorems ........................ 127
5.3. Reduced functions .................................... 129
5.4. The capacity of a compact set ................ .......... 134
5.5. Inner and outer capacity ............................... 137
5.6. Capacitable sets ...................................... 143
5.7. The fundamental convergence theorem .................. 146
5.8. Logarithmic capacity .................................. 150
5.9. Hausdorff measure and capacity ........................ 156
5.10. Exercises ............................................ 159
6. The Dirichlet Problem
6.1. Introduction ................... ...................... 163
6.2. Upper and lower PWB solutions ........................ 164
6.3. Further properties of PWB solutions .................... 167
6.4. Harmonic measure .................................... 172
6.5. Negligible sets ........................................ 177
6.6. Boundary behaviour .................................. 179
6.7. Behaviour near infinity ................................ 188
6.8. Regularity and the Green function ...................... 189
6.9. PWB solutions and reduced functions ................... 191
6.10. Superharmonic extension .............................. 192
6.11. Exercises ............................................ 193
7. The Fine Topology
7.1. Introduction ......................................... 197
7.2. Thin sets ............................................ 199
7.3. Thin sets and reduced functions ........................ 201
7.4. Fine limits ........................................... 206
7.5. Thin sets and the Dirichlet problem ..................... 208
7.6. Thinness at infinity ................................... 214
7.7. Wiener's criterion ..................................... 217
7.8. Limit properties of superharmonic functions .............. 221
7.9. Harmonic approximation .............................. 226
Contents Xl
8. The Martin Boundary
8.1. The Martin kernel and Martin boundary ... ......... ... .. 233
8.2. Reduced functions and minimal harmonic functions ... .... 241
8.3. Reductions on ,10 and ,11 ... ........... ........ .. . . .. . 246
8.4. The Martin representation .......... ..... ....... .... ... 250
8.5. The Martin boundary of a strip .. .. .... ... .. .... .... .. . 252
8.6. The Martin kernel and the Kelvin transform ....... ... .. . 256
8.7. The boundary Harnack principle for Lipschitz domains .... 259
8.8. The Martin boundary of a Lipschitz domain .... ... ...... 269
9. Boundary Limits
9.1. Swept measures and the Dirichlet problem for the Martin
compactification .. ... .. .. ... .. .. ..... .. .. ... .......... 273
9.2. Minimal thinness .. .. .. ... .. .... ..... ...... ....... .. .. 279
9.3. Minimal fine limits ....................... .. ........ .. . 284
9.4. The Fatou-Nalm-Doob theorem .... .. ... ..... ..... ..... 287
9.5. Minimal thinness in sub domains ..... ... .... ..... ...... . 290
9.6. Refinements of limit theorems .. .. ... ..... ........ ... .. . 294
9.7. Minimal thinness in a half-space .. .. ... .. .. ...... ....... 296
Appendix ... ....... ... ... ....... .. ... ...... ... .... .. ... ..... 305
Historical Notes .. ....... ..... ...... ... ....... .............. 309
References ...... ..... .... ..... ..... .... .. .. ..... ........ .... 317
Symbol Index ... .... .... .. ....... .... ....... ... ... ..... ... .. 329
Index ....... ..... ... ... ... ...... ......... ...... ......... .... 331