Potential Theory

by Helms, Lester L.

• ISBN13: 9781848823181
• ISBN10: 1848823185
• Format: Paperback
• Copyright: 2009-06-15
• Publisher: Springer Verlag

Summary

This book presents a clear path from calculus to classical potential theory and beyond with the aim of moving the reader into a fertile area of mathematical research as quickly as possible. The first half of the book develops the subject matter from first principles using only calculus. The second half comprises more advanced material for those with a senior undergraduate or beginning graduate course in real analysis. For specialized regions, solutions of Laplace's equation are constructed having prescribed normal derivatives on the flat portion of the boundary and prescribed values on the remaining portion of the boundary. By means of transformations known as diffeomorphisms, these solutions are morphed into local solutions on regions with curved boundaries. The Perron-Weiner-Brelot method is then used to construct global solutions for elliptic PDEs involving a mixture of prescribed values of a boundary differential operator on part of the boundary and prescribed values on the remainder of the boundary.

Table of Contents

Preliminaries	p. 1
Notation	p. 1
Useful Theorems	p. 4
Laplace's Equation	p. 7
Introduction	p. 7
Green's Theorem	p. 8
Fundamental Harmonic Function	p. 9
The Mean Value Property	p. 10
Poisson Integral Formula	p. 14
Gauss' Averaging Principle	p. 21
The Dirichlet Problem for a Ball	p. 24
Kelvin Transformation	p. 32
Poisson Integral for Half-space	p. 33
Neumann Problem for a Disk	p. 39
Neumann Problem for the Ball	p. 42
Spherical Harmonics	p. 49
The Dirichlet Problem	p. 53
Introduction	p. 53
Sequences of Harmonic Functions	p. 54
Superharmonic Functions	p. 59
Properties of Superharmonic Functions	p. 65
Approximation of Superharmonic Functions	p. 71
Perron-Wiener Method	p. 75
The Radial Limit Theorem	p. 88
Nontangential Boundary Limit Theorem	p. 92
Harmonic Measure	p. 100
Green Functions	p. 107
Introduction	p. 107
Green Functions	p. 107
Symmetry of the Green Functions	p. 115
Green Potentials	p. 122
Riesz Decomposition	p. 135
Properties of Potentials	p. 144
Negligible Sets	p. 149
Introduction	p. 149
Superharmonic Extensions	p. 149
Reduction of Superharmonic Functions	p. 158
Capacity	p. 163
Boundary Behavior of the Green Function	p. 178
Applications	p. 182
Sweeping	p. 189
Dirichlet Problem for Unbounded Regions	p. 197
Introduction	p. 197
Exterior Dirichlet Problem	p. 197
PWB Method for Unbounded Regions	p. 204
Boundary Behavior	p. 210
Intrinsic Topology	p. 223
Thin Sets	p. 224
Thinness and Regularity	p. 228
Energy	p. 241
Introduction	p. 241
Energy Principle	p. 241
Mutual Energy	p. 249
Projections of Measures	p. 257
Wiener's Test	p. 260
Interpolation and Monotonicity	p. 267
Introduction	p. 267
Höet;lder Spaces	p. 268
Global Interpolation	p. 273
Interpolation of Weighted Norms	p. 280
Inner Norms	p. 284
Monotonicity	p. 288
Newtonian Potential	p. 303
Introduction	p. 303
Subnewtonian Kernels	p. 303
Poisson's Equation	p. 316
Höet;lder Continuity of Second Derivatives	p. 319
The Reflection Principle	p. 326
Elliptic Operators	p. 333
Introduction	p. 333
Linear Spaces	p. 334
Constant Coefficients	p. 335
Schauder Interior Estimates	p. 338
Maximum Principles	p. 343
The Dirichlet Problem for a Ball	p. 347
Dirichlet Problem for Bounded Domains	p. 353
Barriers	p. 358
Apriori Bounds	p. 371
Introduction	p. 371
Green Function for a Half-space	p. 372
Mixed Boundary Conditions for Laplacian	p. 376
Nonconstant Coefficients	p. 385
Oblique Derivative Problem	p. 391
Introduction	p. 391
Boundary Maximum Principle	p. 391
Curved Boundaries	p. 405
Superfunctions for Elliptic Operators	p. 410
Regularity of Boundary Points	p. 420
References	p. 431
Index	p. 435
Notation	p. 439
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