Potential Theory (eBook)

Lester Helms (Autor)

eBook Download: PDF

2009 | 1. Auflage
XI, 441 Seiten
Springer London (Verlag)
978-1-84882-319-8 (ISBN)

Infos

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.

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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 "e;. . . When I read a journal article, I often ?nd mistakes. Whether I can ?x them is irrelevant. The literature is unreliable. "e; 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.

Inhalt

Preliminaries.- Laplace's Equation.- The Dirichlet Problem.- Green Functions.- Negligible Sets.- Dirichlet Problem for Unbounded Regions.- Energy.- Interpolation and Monotonicity.- Newtonian Potential.- Elliptic Operators.- Apriori Bounds.- Oblique Derivative Problem.

Produkt-Details

• Erscheint lt. Verlag: 12.6.2009
• Reihe/Serie: Universitext
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Schlagworte: boundary • dirichlet • Elliptic • Mathematics • Oblique • partial differential equation • Physics • Potential • Potential Theory
• ISBN-10: 1-84882-319-3 / 1848823193
• ISBN-13: 978-1-84882-319-8 / 9781848823198