Potential Theory

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Potential Theory

Lester L. Helms
Potential Theory Book PDF

Author : Lester L. Helms
Publisher : Springer Science & Business Media
Page : 494 pages
File Size : 46,6 Mb
Release : 2014-04-10
Category : Mathematics
ISBN : 9781447164227

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Potential Theory by Lester L. Helms Pdf

Potential Theory presents a clear path from calculus to classical potential theory and beyond, with the aim of moving the reader into the area of mathematical research as quickly as possible. The subject matter is developed from first principles using only calculus. Commencing with the inverse square law for gravitational and electromagnetic forces and the divergence theorem, the author develops methods for constructing solutions of Laplace's equation on a region with prescribed values on the boundary of the region. The latter half of the book addresses more advanced material aimed at those with the background of a senior undergraduate or beginning graduate course in real analysis. Starting with solutions of the Dirichlet problem subject to mixed boundary conditions on the simplest of regions, methods of morphing such solutions onto solutions of Poisson's equation on more general regions are developed using diffeomorphisms and the Perron-Wiener-Brelot method, culminating in application to Brownian motion. In this new edition, many exercises have been added to reconnect the subject matter to the physical sciences. This book will undoubtedly be useful to graduate students and researchers in mathematics, physics and engineering.

Function Spaces and Potential Theory

David R. Adams,Lars I. Hedberg
Function Spaces and Potential Theory Book PDF

Author : David R. Adams,Lars I. Hedberg
Publisher : Springer Science & Business Media
Page : 372 pages
File Size : 42,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783662032824

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Function Spaces and Potential Theory by David R. Adams,Lars I. Hedberg Pdf

"..carefully and thoughtfully written and prepared with, in my opinion, just the right amount of detail included...will certainly be a primary source that I shall turn to." Proceedings of the Edinburgh Mathematical Society

Foundations of Potential Theory

Oliver Dimon Kellogg
Foundations of Potential Theory Book PDF

Author : Oliver Dimon Kellogg
Publisher : Read Books Ltd
Page : 396 pages
File Size : 55,9 Mb
Release : 2011-03-23
Category : Philosophy
ISBN : 9781446547830

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Foundations of Potential Theory by Oliver Dimon Kellogg Pdf

The present volume gives a systematic treatment of potential functions. It takes its origin in two courses, one elementary and one advanced, which the author has given at intervals during the last ten years, and has a two-fold purpose first, to serve as an introduction for students whose attainments in the Calculus include some knowledge of partial derivatives and multiple and line integrals and secondly, to provide the reader with the fundamentals of the subject, so that he may proceed immediately to the applications, or to - the periodical literature of the day. It is inherent in the nature of the subject that physical intuition and illustration be appealed to freely, and this has been done. However, in order that the ok may present sound ideals to the student, and also serve the mathematician, both for purposes of reference and as a basis for further developments, the proofs have been given by rigorous methods. This has led, at a number of points, to results either not found elsewhere, or not readily accessible. Thus, Chapter IV contains a proof for the general regular region of the divergence theorem Gauss, or Greens theorem on the reduction of volume to surface integrals. The treatment of the fundamental existence theorems in Chapter XI by means of integral equations meets squarely the difficulties incident to the discontinuity of the kernel, and the same chapter gives an account of the most recent developments with respect to the Pirichlet problem. Exercises are introduced in the conviction that no mastery of a mathematical subject is possible without working with it. They are designed primarily to illustrate or extend the theory, although the desirability of requiring an occasional concrete numerical result has not been lost sight of.

Classical Potential Theory

David H. Armitage,Stephen J. Gardiner
Classical Potential Theory Book PDF

Author : David H. Armitage,Stephen J. Gardiner
Publisher : Springer Science & Business Media
Page : 343 pages
File Size : 43,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781447102335

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Classical Potential Theory by David H. Armitage,Stephen J. Gardiner Pdf

A long-awaited, updated introductory text by the world leaders in potential theory. This essential reference work covers all aspects of this major field of mathematical research, from basic theory and exercises to more advanced topological ideas. The largely self-contained presentation makes it basically accessible to graduate students.

Potential Theory

John Wermer
Potential Theory Book PDF

Author : John Wermer
Publisher : Springer Science & Business Media
Page : 156 pages
File Size : 43,9 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9783662127278

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Potential Theory by John Wermer Pdf

Potential theory grew out of mathematical physics, in particular out of the theory of gravitation and the theory of electrostatics. Mathematical physicists such as Poisson and Green introduced some of the central ideas of the subject. A mathematician with a general knowledge of analysis may find it useful to begin his study of classical potential theory by looking at its physical origins. Sections 2, 5 and 6 of these Notes give in part heuristic arguments based on physical considerations. These heuristic arguments suggest mathematical theorems and provide the mathematician with the problem of finding the proper hypotheses and mathematical proofs. These Notes are based on a one-semester course given by the author at Brown University in 1971. On the part of the reader, they assume a knowledge of Real Function Theory to the extent of a first year graduate course. In addition some elementary facts regarding harmonic functions are aS$umed as known. For convenience we have listed these facts in the Appendix. Some notation is also explained there. Essentially all the proofs we give in the Notes are for Euclidean 3-space R3 and Newtonian potentials ~.

Potential Theory on Locally Compact Abelian Groups

C. van den Berg,G. Forst
Potential Theory on Locally Compact Abelian Groups Book PDF

Author : C. van den Berg,G. Forst
Publisher : Springer Science & Business Media
Page : 205 pages
File Size : 51,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642661280

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Potential Theory on Locally Compact Abelian Groups by C. van den Berg,G. Forst Pdf

Classical potential theory can be roughly characterized as the study of Newtonian potentials and the Laplace operator on the Euclidean space JR3. It was discovered around 1930 that there is a profound connection between classical potential 3 theory and the theory of Brownian motion in JR . The Brownian motion is determined by its semigroup of transition probabilities, the Brownian semigroup, and the connection between classical potential theory and the theory of Brownian motion can be described analytically in the following way: The Laplace operator is the infinitesimal generator for the Brownian semigroup and the Newtonian potential kernel is the" integral" of the Brownian semigroup with respect to time. This connection between classical potential theory and the theory of Brownian motion led Hunt (cf. Hunt [2]) to consider general "potential theories" defined in terms of certain stochastic processes or equivalently in terms of certain semi groups of operators on spaces of functions. The purpose of the present exposition is to study such general potential theories where the following aspects of classical potential theory are preserved: (i) The theory is defined on a locally compact abelian group. (ii) The theory is translation invariant in the sense that any translate of a potential or a harmonic function is again a potential, respectively a harmonic function; this property of classical potential theory can also be expressed by saying that the Laplace operator is a differential operator with constant co efficients.

Classical Potential Theory and Its Probabilistic Counterpart

J. L. Doob
Classical Potential Theory and Its Probabilistic Counterpart Book PDF

Author : J. L. Doob
Publisher : Springer Science & Business Media
Page : 865 pages
File Size : 54,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461252085

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Classical Potential Theory and Its Probabilistic Counterpart by J. L. Doob Pdf

Potential theory and certain aspects of probability theory are intimately related, perhaps most obviously in that the transition function determining a Markov process can be used to define the Green function of a potential theory. Thus it is possible to define and develop many potential theoretic concepts probabilistically, a procedure potential theorists observe withjaun diced eyes in view of the fact that now as in the past their subject provides the motivation for much of Markov process theory. However that may be it is clear that certain concepts in potential theory correspond closely to concepts in probability theory, specifically to concepts in martingale theory. For example, superharmonic functions correspond to supermartingales. More specifically: the Fatou type boundary limit theorems in potential theory correspond to supermartingale convergence theorems; the limit properties of monotone sequences of superharmonic functions correspond surprisingly closely to limit properties of monotone sequences of super martingales; certain positive superharmonic functions [supermartingales] are called "potentials," have associated measures in their respective theories and are subject to domination principles (inequalities) involving the supports of those measures; in each theory there is a reduction operation whose properties are the same in the two theories and these reductions induce sweeping (balayage) of the measures associated with potentials, and so on.

Potential Theory

Jürgen Bliedtner,Wolfhard Hansen
Potential Theory Book PDF

Author : Jürgen Bliedtner,Wolfhard Hansen
Publisher : Springer Science & Business Media
Page : 448 pages
File Size : 51,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642711312

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Potential Theory by Jürgen Bliedtner,Wolfhard Hansen Pdf

During the last thirty years potential theory has undergone a rapid development, much of which can still only be found in the original papers. This book deals with one part of this development, and has two aims. The first is to give a comprehensive account of the close connection between analytic and probabilistic potential theory with the notion of a balayage space appearing as a natural link. The second aim is to demonstrate the fundamental importance of this concept by using it to give a straight presentation of balayage theory which in turn is then applied to the Dirichlet problem. We have considered it to be beyond the scope of this book to treat further topics such as duality, ideal boundary and integral representation, energy and Dirichlet forms. The subject matter of this book originates in the relation between classical potential theory and the theory of Brownian motion. Both theories are linked with the Laplace operator. However, the deep connection between these two theories was first revealed in the papers of S. KAKUTANI [1], [2], [3], M. KAC [1] and J. L. DO DB [2] during the period 1944-54: This can be expressed by the·fact that the harmonic measures which occur in the solution of the Dirichlet problem are hitting distri butions for Brownian motion or, equivalently, that the positive hyperharmonic func tions for the Laplace equation are the excessive functions of the Brownian semi group.

Potential Theory in Gravity and Magnetic Applications

Richard J. Blakely
Potential Theory in Gravity and Magnetic Applications Book PDF

Author : Richard J. Blakely
Publisher : Cambridge University Press
Page : 468 pages
File Size : 53,9 Mb
Release : 1996-09-13
Category : Mathematics
ISBN : 0521575478

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Potential Theory in Gravity and Magnetic Applications by Richard J. Blakely Pdf

This text bridges the gap between the classic texts on potential theory and modern books on applied geophysics. It opens with an introduction to potential theory, emphasising those aspects particularly important to earth scientists, such as Laplace's equation, Newtonian potential, magnetic and electrostatic fields, and conduction of heat. The theory is then applied to the interpretation of gravity and magnetic anomalies, drawing on examples from modern geophysical literature. Topics explored include regional and global fields, forward modeling, inverse methods, depth-to-source estimation, ideal bodies, analytical continuation, and spectral analysis. The book includes numerous exercises and a variety of computer subroutines written in FORTRAN. Graduate students and researchers in geophysics will find this book essential.

Nonlinear Potential Theory of Degenerate Elliptic Equations

Juha Heinonen,Tero Kipelainen,Olli Martio
Nonlinear Potential Theory of Degenerate Elliptic Equations Book PDF

Author : Juha Heinonen,Tero Kipelainen,Olli Martio
Publisher : Courier Dover Publications
Page : 417 pages
File Size : 45,5 Mb
Release : 2018-05-16
Category : Mathematics
ISBN : 9780486824253

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Nonlinear Potential Theory of Degenerate Elliptic Equations by Juha Heinonen,Tero Kipelainen,Olli Martio Pdf

A self-contained treatment appropriate for advanced undergraduate and graduate students, this volume offers a detailed development of the necessary background for its survey of the nonlinear potential theory of superharmonic functions. Starting with the theory of weighted Sobolev spaces, the text advances to the theory of weighted variational capacity. Succeeding chapters investigate solutions and supersolutions of equations, with emphasis on refined Sobolev spaces, variational integrals, and harmonic functions. Chapter 7 defines superharmonic functions via the comparison principle, and chapters 8 through 14 form the core of the nonlinear potential theory of superharmonic functions. Topics include balayage; Perron's method, barriers, and resolutivity; polar sets; harmonic measure; fine topology; harmonic morphisms; and quasiregular mappings. The book concludes with explorations of axiomatic nonlinear potential theory and helpful appendixes.

Nonlinear Potential Theory and Weighted Sobolev Spaces

Bengt O. Turesson
Nonlinear Potential Theory and Weighted Sobolev Spaces Book PDF

Author : Bengt O. Turesson
Publisher : Springer
Page : 188 pages
File Size : 52,8 Mb
Release : 2007-05-06
Category : Mathematics
ISBN : 9783540451686

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Nonlinear Potential Theory and Weighted Sobolev Spaces by Bengt O. Turesson Pdf

The book systematically develops the nonlinear potential theory connected with the weighted Sobolev spaces, where the weight usually belongs to Muckenhoupt's class of Ap weights. These spaces occur as solutions spaces for degenerate elliptic partial differential equations. The Sobolev space theory covers results concerning approximation, extension, and interpolation, Sobolev and Poincaré inequalities, Maz'ya type embedding theorems, and isoperimetric inequalities. In the chapter devoted to potential theory, several weighted capacities are investigated. Moreover, "Kellogg lemmas" are established for various concepts of thinness. Applications of potential theory to weighted Sobolev spaces include quasi continuity of Sobolev functions, Poincaré inequalities, and spectral synthesis theorems.

Potential Theory in Applied Geophysics

Kalyan Kumar Roy
Potential Theory in Applied Geophysics Book PDF

Author : Kalyan Kumar Roy
Publisher : Springer Science & Business Media
Page : 661 pages
File Size : 41,9 Mb
Release : 2007-11-15
Category : Science
ISBN : 9783540723349

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Potential Theory in Applied Geophysics by Kalyan Kumar Roy Pdf

This book introduces the principles of gravitational, magnetic, electrostatic, direct current electrical and electromagnetic fields, with detailed solutions of Laplace and electromagnetic wave equations by the method of separation of variables. Discussion includes behaviours of the scalar and vector potential and the nature of the solutions of these boundary value problems, along with the use of complex variables and conformal transformation, Green's theorem, Green's formula and Green's functions.

Multiscale Potential Theory

Willi Freeden,Volker Michel
Multiscale Potential Theory Book PDF

Author : Willi Freeden,Volker Michel
Publisher : Springer Science & Business Media
Page : 522 pages
File Size : 40,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461220480

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Multiscale Potential Theory by Willi Freeden,Volker Michel Pdf

This self-contained text/reference provides a basic foundation for practitioners, researchers, and students interested in any of the diverse areas of multiscale (geo)potential theory. New mathematical methods are developed enabling the gravitational potential of a planetary body to be modeled using a continuous flow of observations from land or satellite devices. Harmonic wavelets methods are introduced, as well as fast computational schemes and various numerical test examples. Presented are multiscale approaches for numerous geoscientific problems, including geoidal determination, magnetic field reconstruction, deformation analysis, and density variation modelling With exercises at the end of each chapter, the book may be used as a textbook for graduate-level courses in geomathematics, applied mathematics, and geophysics. The work is also an up-to-date reference text for geoscientists, applied mathematicians, and engineers.

Potential Theory in the Complex Plane

Thomas Ransford
Potential Theory in the Complex Plane Book PDF

Author : Thomas Ransford
Publisher : Cambridge University Press
Page : 246 pages
File Size : 52,6 Mb
Release : 1995-03-16
Category : Mathematics
ISBN : 0521466547

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Potential Theory in the Complex Plane by Thomas Ransford Pdf

Potential theory is the broad area of mathematical analysis encompassing such topics as harmonic and subharmonic functions.

Potential Theory and Dynamics on the Berkovich Projective Line

Matthew Baker,Robert S. Rumely
Potential Theory and Dynamics on the Berkovich Projective Line Book PDF

Author : Matthew Baker,Robert S. Rumely
Publisher : American Mathematical Soc.
Page : 466 pages
File Size : 48,9 Mb
Release : 2010-03-10
Category : Mathematics
ISBN : 9780821849248

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Potential Theory and Dynamics on the Berkovich Projective Line by Matthew Baker,Robert S. Rumely Pdf

The purpose of this book is to develop the foundations of potential theory and rational dynamics on the Berkovich projective line over an arbitrary complete, algebraically closed non-Archimedean field. In addition to providing a concrete and ``elementary'' introduction to Berkovich analytic spaces and to potential theory and rational iteration on the Berkovich line, the book contains applications to arithmetic geometry and arithmetic dynamics. A number of results in the book are new, and most have not previously appeared in book form. Three appendices--on analysis, $\mathbb{R}$-trees, and Berkovich's general theory of analytic spaces--are included to make the book as self-contained as possible. The authors first give a detailed description of the topological structure of the Berkovich projective line and then introduce the Hsia kernel, the fundamental kernel for potential theory. Using the theory of metrized graphs, they define a Laplacian operator on the Berkovich line and construct theories of capacities, harmonic and subharmonic functions, and Green's functions, all of which are strikingly similar to their classical complex counterparts. After developing a theory of multiplicities for rational functions, they give applications to non-Archimedean dynamics, including local and global equidistribution theorems, fixed point theorems, and Berkovich space analogues of many fundamental results from the classical Fatou-Julia theory of rational iteration. They illustrate the theory with concrete examples and exposit Rivera-Letelier's results concerning rational dynamics over the field of $p$-adic complex numbers. They also establish Berkovich space versions of arithmetic results such as the Fekete-Szego theorem and Bilu's equidistribution theorem.