Potential Theory
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Beschreibung
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.
Written by the author of Introduction to Potential Theory, this is a new and modern textbook that introduces all the important concepts of classical potential theory Equips readers for further study in elliptic partial differential equations, axiomatic potential theory, and the interplay between probability theory and potential theory Includes supplementary material: sn.pub/extras
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Lester Helms
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Boundary Dirichlet Elliptic Oblique Potential Potential theory mathematics partial differential equation physics partial differential equations
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From the reviews:
“The author sets the goal of the book as getting the reader from real analysis to the front line of potential theory as quickly as possible. … Each chapter begins with some physical and historical context … . The excellent index is also very helpful in navigating the material. … a fine text for self-study, reference, or a graduate course. Researchers in the field will consider it a standard and those in an adjacent field … will also find it a valuable reference.” (Bill Wood, The Mathematical Association of America, May, 2010)
“The first part of the book deals with the basics of classical potential theory while the rest of the book deals with the solution to elliptic partial differential equations with various boundary conditions. … Proofs are given in easy to follow detail. … the book is very suitable as a textbook … . On the whole, this book is a very useful addition to available resources. … There is also an index, a notation guide and an extensive bibliography.” (P. Lappan, Mathematical Reviews, Issue 2011 a)
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