Linear Holomorphic Partial Differential Equations and Classical Potential Theory

Linear Holomorphic Partial Differential Equations and Classical Potential Theory PDF

Author: Dmitry Khavinson

Publisher: American Mathematical Soc.

Published: 2018-07-09

Total Pages: 214

ISBN-13: 1470437805

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Why do solutions of linear analytic PDE suddenly break down? What is the source of these mysterious singularities, and how do they propagate? Is there a mean value property for harmonic functions in ellipsoids similar to that for balls? Is there a reflection principle for harmonic functions in higher dimensions similar to the Schwarz reflection principle in the plane? How far outside of their natural domains can solutions of the Dirichlet problem be extended? Where do the continued solutions become singular and why? This book invites graduate students and young analysts to explore these and many other intriguing questions that lead to beautiful results illustrating a nice interplay between parts of modern analysis and themes in “physical” mathematics of the nineteenth century. To make the book accessible to a wide audience including students, the authors do not assume expertise in the theory of holomorphic PDE, and most of the book is accessible to anyone familiar with multivariable calculus and some basics in complex analysis and differential equations.

Potential Theory, Surveys and Problems

Potential Theory, Surveys and Problems PDF

Author: Josef Kral

Publisher: Springer

Published: 2007-02-08

Total Pages: 276

ISBN-13: 3540459529

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The volume comprises eleven survey papers based on survey lectures delivered at the Conference in Prague in July 1987, which covered various facets of potential theory, including its applications in other areas. The survey papers deal with both classical and abstract potential theory and its relations to partial differential equations, stochastic processes and other branches such as numerical analysis and topology. A collection of problems from potential theory, compiled on the occasion of the conference, is included, with additional commentaries, in the second part of this volume.

Potential Theory

Potential Theory PDF

Author: Lester Helms

Publisher: Springer Science & Business Media

Published: 2009-05-27

Total Pages: 442

ISBN-13: 1848823193

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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 “. . . 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.

Linear and Quasilinear Parabolic Systems: Sobolev Space Theory

Linear and Quasilinear Parabolic Systems: Sobolev Space Theory PDF

Author: David Hoff

Publisher: American Mathematical Soc.

Published: 2020-11-18

Total Pages: 226

ISBN-13: 1470461617

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This monograph presents a systematic theory of weak solutions in Hilbert-Sobolev spaces of initial-boundary value problems for parabolic systems of partial differential equations with general essential and natural boundary conditions and minimal hypotheses on coefficients. Applications to quasilinear systems are given, including local existence for large data, global existence near an attractor, the Leray and Hopf theorems for the Navier-Stokes equations and results concerning invariant regions. Supplementary material is provided, including a self-contained treatment of the calculus of Sobolev functions on the boundaries of Lipschitz domains and a thorough discussion of measurability considerations for elements of Bochner-Sobolev spaces. This book will be particularly useful both for researchers requiring accessible and broadly applicable formulations of standard results as well as for students preparing for research in applied analysis. Readers should be familiar with the basic facts of measure theory and functional analysis, including weak derivatives and Sobolev spaces. Prior work in partial differential equations is helpful but not required.

Research Problems in Function Theory

Research Problems in Function Theory PDF

Author: Walter K. Hayman

Publisher: Springer Nature

Published: 2019-09-07

Total Pages: 284

ISBN-13: 3030251659

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In 1967 Walter K. Hayman published ‘Research Problems in Function Theory’, a list of 141 problems in seven areas of function theory. In the decades following, this list was extended to include two additional areas of complex analysis, updates on progress in solving existing problems, and over 520 research problems from mathematicians worldwide. It became known as ‘Hayman's List’. This Fiftieth Anniversary Edition contains the complete ‘Hayman's List’ for the first time in book form, along with 31 new problems by leading international mathematicians. This list has directed complex analysis research for the last half-century, and the new edition will help guide future research in the subject. The book contains up-to-date information on each problem, gathered from the international mathematics community, and where possible suggests directions for further investigation. Aimed at both early career and established researchers, this book provides the key problems and results needed to progress in the most important research questions in complex analysis, and documents the developments of the past 50 years.

Notions of Convexity

Notions of Convexity PDF

Author: Lars Hörmander

Publisher: Springer Science & Business Media

Published: 2007-06-25

Total Pages: 424

ISBN-13: 0817645853

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The first two chapters of this book are devoted to convexity in the classical sense, for functions of one and several real variables respectively. This gives a background for the study in the following chapters of related notions which occur in the theory of linear partial differential equations and complex analysis such as (pluri-)subharmonic functions, pseudoconvex sets, and sets which are convex for supports or singular supports with respect to a differential operator. In addition, the convexity conditions which are relevant for local or global existence of holomorphic differential equations are discussed.

Potential Theory

Potential Theory PDF

Author: Lester L. Helms

Publisher: Springer Science & Business Media

Published: 2014-04-10

Total Pages: 494

ISBN-13: 1447164229

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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.

Virtual Fundamental Cycles in Symplectic Topology

Virtual Fundamental Cycles in Symplectic Topology PDF

Author: John W. Morgan

Publisher: American Mathematical Soc.

Published: 2019-04-12

Total Pages: 300

ISBN-13: 1470450143

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The method of using the moduli space of pseudo-holomorphic curves on a symplectic manifold was introduced by Mikhail Gromov in 1985. From the appearance of Gromov's original paper until today this approach has been the most important tool in global symplectic geometry. To produce numerical invariants of these manifolds using this method requires constructing a fundamental cycle associated with moduli spaces. This volume brings together three approaches to constructing the “virtual” fundamental cycle for the moduli space of pseudo-holomorphic curves. All approaches are based on the idea of local Kuranishi charts for the moduli space. Workers in the field will get a comprehensive understanding of the details of these constructions and the assumptions under which they can be made. These techniques and results will be essential in further applications of this approach to producing invariants of symplectic manifolds.

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