Notes on classical potential theory

264 Pages·2004·1.268 MB·English

by  Papadimitrakis M.

NOTES ON
CLASSICAL
POTENTIAL THEORY
M. Papadimitrakis
Department of Mathematics
University of Crete
January 2004
2
Forword
During the fall semester of the academic year 1990-1991 I gave a course on
Classical Potential Theory attended by an excellent class of graduate students
oftheDepartmentofMathematicsofWashingtonUniversity. Thatwasmyfirst
time to teach such a course and, I have to say, besides sporadic knowledge of a
fewfacts directlyrelatedto complexanalysis,Ihadnoseriousknowledgeofthe
subject. The result was: many sleepless nights reading books, trying to choose
thematerialtobepresentedandpreparinghand-writtennotesforthestudents.
The books I found very useful and which determined the choice of material
were the superb “E´l´ements de la Th´eorie Classique du Potentiel” by M. Brelot
andthe“SelectedProblemsonExceptionalSets”byL.Carleson. Othersources
were: “Some Topics in the Theory of Functions of One Complex Variable” by
W.Fuchs,unpublishednotes on“HarmonicMeasures”byJ.Garnett,“Subhar-
monic Functions” by W. Hayman and P. Kennedy, “Introduction to Potential
Theory” by L. Helms, “Foundations of Modern Potential Theory” by N. Land-
kof, “Subharmonic Functions” by T. Rado and “Potential Theory in Modern
Function Theory” by M. Tsuji.
This is a slightly expanded version of the original notes with very few
changes. The principle has remained the same, namely to present an overview
of the classical theory at the level of a graduate course. The part called “Pre-
liminaries” is new and its contents were silently taken for granted during the
originalcourse. ThemainmaterialistheDivergenceTheoremandGreen’sFor-
mula, a short course on holomorphic functions (, since their real parts are the
main examples of harmonic functions in the plane and, also, since one of the
centralresultsistheproofoftheRiemannMappingTheoremthroughpotential
theory), some basic facts about semi-continuous functions and very few ele-
mentary results about distributions and the Fourier transform. Except for the
Divergence Theorem,the Arzela-AscoliTheorem, the Radon-Riesz Representa-
tion Theorem and, of course, the basic facts of measure theory and functional
analysis,all ofwhichare usedbut notprovedhere, allothermaterialcontained
in these notes is proved with sufficient detail.
Material which was not included in the original notes: the section on har-
monic conjugates in the first chapter (it, actually, contains a new proof of the
existence of a harmonic conjugate in a simply-connected subset of the plane);
the section on the differentiability of potentials in the second chapter; the sec-
tions on superharmonic functions at and on Poisson integrals at in the
∞ ∞
fourth chapter; an additional proof of the result about the direct connection
betweenGreen’sfunctionandharmonicmeasureinthefifthchapter(indicating
the role of the normal derivative of Green’s function as an approximation to
the identity); the subadditivity of capacityin the eigth chapter;the sections on
polar sets and thin sets in the ninth chapter. The definition of the notion of
“quasi-almosteverywhere”inthe eigthchapterhasbeen changed. The proofof
the Riemann Mapping Theorem in the ninth chapter is corrected and given in
3
full detail, not relying on “obvious” topological facts any more. In the original
coursethe proof(taken fromthe notes of J. Garnett) of Wiener’s Theoremwas
presented only in dimension 2. Now, the proof is given in all dimensions.
A short and very classical application of potential theory in dimension 1 on
the convergence of trigonometric series is missing from this set of notes, since
it is quite specialized. What is, also, missing is a short chapter on the metrical
properties of capacity and an example of a Cantor-like set. But this will be
includedvery soon, afterit is expanded as achapter on “Capacity, capacitability
and Hausdorff measures”.
Besidesthe newmaterial,thereis are-organizationwhichresults,Ihope,to
better exposition.
Here, I would like to thank the Department of Mathematics of Washington
University for giving me the opportunity to teach the original course and the
graduate class which attended it with great care and enthusiasm.
4
Contents
I Preliminaries 9
0.1 Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
0.2 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
0.3 Holomorphic Functions. . . . . . . . . . . . . . . . . . . . . . . . 18
0.4 Equicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
0.5 Semi-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
0.6 Borel Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
0.7 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
0.8 Concavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
0.9 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . 51
II Main Theory 65
1 Harmonic Functions 67
1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
1.2 Maximum-minimum principle . . . . . . . . . . . . . . . . . . . . 69
1.3 Differentiability of harmonic functions . . . . . . . . . . . . . . . 70
1.4 Holomorphy and harmonic conjugates . . . . . . . . . . . . . . . 71
1.5 Fundamental solution . . . . . . . . . . . . . . . . . . . . . . . . 75
1.6 Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
1.7 Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
1.8 The representation formula . . . . . . . . . . . . . . . . . . . . . 80
1.9 Poisson integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
1.10 Consequences of the Poisson formula . . . . . . . . . . . . . . . . 85
1.11 Monotone sequences . . . . . . . . . . . . . . . . . . . . . . . . . 90
1.12 Normal families of harmonic functions . . . . . . . . . . . . . . . 91
1.13 Harmonic distributions . . . . . . . . . . . . . . . . . . . . . . . . 93
2 Superharmonic Functions 97
2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
2.2 Minimum principle . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.3 Blaschke-Privaloffparameters . . . . . . . . . . . . . . . . . . . . 100
2.4 Poisson modification . . . . . . . . . . . . . . . . . . . . . . . . . 105
2.5 Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5
6 CONTENTS
2.6 Differentiability of potentials . . . . . . . . . . . . . . . . . . . . 109
2.7 Approximation, properties of means . . . . . . . . . . . . . . . . 112
2.8 The Perronprocess . . . . . . . . . . . . . . . . . . . . . . . . . . 116
2.9 The largest harmonic minorant . . . . . . . . . . . . . . . . . . . 117
2.10 Superharmonic distributions . . . . . . . . . . . . . . . . . . . . . 119
2.11 The theorem of F. Riesz . . . . . . . . . . . . . . . . . . . . . . . 122
2.12 Derivatives of superharmonic functions . . . . . . . . . . . . . . . 123
3 The Problem of Dirichlet 125
3.1 The generalized solution . . . . . . . . . . . . . . . . . . . . . . . 125
3.2 Properties of the generalized solution . . . . . . . . . . . . . . . . 128
3.3 Wiener’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.4 Harmonic measure . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.5 Sets of zero harmonic measure . . . . . . . . . . . . . . . . . . . 138
3.6 Barriers and regularity . . . . . . . . . . . . . . . . . . . . . . . . 140
3.7 Regularity and the problem of Dirichlet . . . . . . . . . . . . . . 143
3.8 Criteria for regularity . . . . . . . . . . . . . . . . . . . . . . . . 145
4 The Kelvin Transform 149
4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.2 Harmonic functions at . . . . . . . . . . . . . . . . . . . . . . 150
∞
4.3 Superharmonic functions at . . . . . . . . . . . . . . . . . . . 154
∞
4.4 Poisson integrals at . . . . . . . . . . . . . . . . . . . . . . . . 155
∞
4.5 The effect of the dimension . . . . . . . . . . . . . . . . . . . . . 156
4.6 Dimension 2, in particular . . . . . . . . . . . . . . . . . . . . . . 157
5 Green’s Function 161
5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.2 Green’s function, the problem of Dirichlet and harmonic measure 162
5.3 A few examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.4 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.5 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.6 Green’s function and regularity . . . . . . . . . . . . . . . . . . . 166
5.7 Extensions of Green’s Function . . . . . . . . . . . . . . . . . . . 167
5.8 Green’s potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.9 The Decomposition Theorem of F. Riesz . . . . . . . . . . . . . . 172
5.10 Green’s function and harmonic measure . . . . . . . . . . . . . . 173
5.11 as interior point. Mainly, n=2 . . . . . . . . . . . . . . . . . 183
∞
6 Potentials 187
6.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.2 Potentials of non-negative Borel measures . . . . . . . . . . . . . 189
6.3 The maximum principle for potentials . . . . . . . . . . . . . . . 192
6.4 The continuity principle for potentials . . . . . . . . . . . . . . . 195
CONTENTS 7
7 Energy 197
7.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.2 Representation of energy: Green’s kernel . . . . . . . . . . . . . . 199
7.3 Measures of finite energy: Green’s kernel . . . . . . . . . . . . . . 205
7.4 Representation of energy: kernels of first type . . . . . . . . . . . 206
7.5 Measures of finite energy: kernels of first type . . . . . . . . . . . 212
8 Capacity 217
8.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
8.2 Equilibrium measures . . . . . . . . . . . . . . . . . . . . . . . . 221
8.3 Transfinite diameter . . . . . . . . . . . . . . . . . . . . . . . . . 229
8.4 The Theorem of Evans . . . . . . . . . . . . . . . . . . . . . . . . 232
8.5 Kernels of variable sign . . . . . . . . . . . . . . . . . . . . . . . 234
9 The Classical Kernels 237
9.1 Extension through sets of zero capacity . . . . . . . . . . . . . . 237
9.2 Sets of zero harmonic measure . . . . . . . . . . . . . . . . . . . 238
9.3 The set of irregular boundary points . . . . . . . . . . . . . . . . 239
9.4 The support of the equilibrium measure . . . . . . . . . . . . . . 243
9.5 Capacity and conformal mapping . . . . . . . . . . . . . . . . . . 244
9.6 Capacity and Green’s function in R2 . . . . . . . . . . . . . . . . 249
9.7 Polar sets and the Theorem of Evans . . . . . . . . . . . . . . . . 251
9.8 The theorem of Wiener . . . . . . . . . . . . . . . . . . . . . . . 255
9.9 Thin Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
8 CONTENTS
Part I
Preliminaries
9