Potential Theory (Universitext)

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. Cover ......Page 1 Universitext......Page 2 Potential Theory......Page 4 1848823185......Page 5 Preface ......Page 8 Contents ......Page 10 0.1 Notation ......Page 14 0.2 Useful Theorems ......Page 17 1.1 Introduction ......Page 20 1.2 Green's Theorem ......Page 21 1.3 Fundamental Harmonic Function ......Page 22 1.4 The Mean Value Property ......Page 23 1.5 Poisson Integral Formula ......Page 27 1.6 Gauss' Averaging Principle ......Page 34 1.7 The Dirichlet Problem for a Ball ......Page 37 1.8 Kelvin Transformation ......Page 45 1.9 Poisson Integral for Half-space ......Page 46 1.10 Neumann Problem for a Disk ......Page 52 1.11 Neumann Problem for the Ball ......Page 55 1.12 Spherical Harmonics ......Page 62 2.1 Introduction ......Page 66 2.2 Sequences of Harmonic Functions ......Page 67 2.3 Superharmonic Functions ......Page 72 2.4 Properties of Superharmonic Functions ......Page 78 2.5 Approximation of Superharmonic Functions ......Page 84 2.6 Perron-Wiener Method ......Page 88 2.7 The Radial Limit Theorem ......Page 101 2.8 Nontangential Boundary Limit Theorem ......Page 105 2.9 Harmonic Measure ......Page 113 3.2 Green Functions ......Page 120 3.3 Symmetry of the Green Function ......Page 128 3.4 Green Potentials ......Page 135 3.5 Riesz Decomposition ......Page 148 3.6 Properties of Potentials ......Page 157 4.2 Superharmonic Extensions ......Page 162 4.3 Reduction of Superharmonic Functions ......Page 171 4.4 Capacity ......Page 176 4.5 Boundary Behavior of the Green Function ......Page 191 4.6 Applications ......Page 195 4.7 Sweeping ......Page 202 5.2 Exterior Dirichlet Problem ......Page 210 5.3 PWB Method for Unbounded Regions ......Page 217 5.4 Boundary Behavior ......Page 223 5.5 Intrinsic Topology ......Page 236 5.6 Thin Sets ......Page 237 5.7 Thinness and Regularity ......Page 241 6.2 Energy Principle ......Page 254 6.3 Mutual Energy ......Page 262 6.4 Projections of Measures ......Page 270 6.5 Wiener's Test ......Page 273 7.1 Introduction ......Page 280 7.2 Hölder Spaces ......Page 281 7.3 Global Interpolation ......Page 286 7.4 Interpolation of Weighted Norms ......Page 293 7.5 Inner Norms ......Page 297 7.6 Monotonicity ......Page 301 8.2 Subnewtonian Kernels ......Page 316 8.3 Poisson's Equation ......Page 329 8.4 Hölder Continuity of Second Derivatives ......Page 332 8.5 The Reflection Principle ......Page 339 9.1 Introduction ......Page 346 9.2 Linear Spaces ......Page 347 9.3 Constant Coefficients ......Page 348 9.4 Schauder Interior Estimates ......Page 351 9.5 Maximum Principles ......Page 356 9.6 The Dirichlet Problem for a Ball ......Page 360 9.7 Dirichlet Problem for Bounded Domains ......Page 366 9.8 Barriers ......Page 371 10.1 Introduction ......Page 384 10.2 Green Function for a Half-space ......Page 385 10.3 Mixed Boundary Conditions for Laplacian ......Page 389 10.4 Nonconstant Coefficients ......Page 398 11.2 Boundary Maximum Principle ......Page 404 11.3 Curved Boundaries ......Page 418 11.4 Superfunctions for Elliptic Operators ......Page 423 11.5 Regularity of Boundary Points ......Page 433 References ......Page 444 Index ......Page 448 Notation ......Page 452 Aimed at graduate students and researchers in mathematics, physics, and engineering, this book presents a clear path from calculus to classical potential theory and beyond, moving the reader into a fertile area of mathematical research as quickly as possible. The author revises and updates material from his classic work, Introduction to Potential Theory (1969), to provide a modern text that introduces all the important concepts of classical potential theory. In the first half of the book, the subject matter is developed meticulously from first principles using only calculus. Commencing with the inverse square law for gravitational and electromagnetic forces and the divergence theorem of the calculus, the author develops methods for constructing solutions of Laplace's equation on a region with prescribed values on the boundary of the region. The second half addresses more advanced material aimed at those with a background of a senior undergraduate or beginning graduate course in real analysis. For specialized regions, namely spherical chips, 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 partial differential equations 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 The?rst six chapters of this book are revised versions of the same chapters in the author's 1969 book, Introduction to Potential Theory. Atthetimeof the writing of that book, I had access to excellent articles,books, and lecture 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 n- comer to the subject will?nd the journal literature to be a maze of excellent 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 Am- ican Mathematical Society, M. Nathanson of Lehman College (CUNY) and (CUNY) Graduate Center said it best “... When I read a journal article, I often?nd mistakes. Whether I can?x them is irrelevant. The literature is unreliable. ” From time to time, someone must try to?nd a path through the maze. In planning this book, it became apparent that a de?ciency 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.