[PDF] Fine Regularity Of Solutions Of Elliptic Partial Differential Equations eBook

Author : Jan Malý
Publisher : American Mathematical Soc.
Page : 309 pages
File Size : 35,93 MB
Release : 1997
Category : Mathematics
ISBN : 0821803352

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The primary objective of this monograph is to give a comprehensive exposition of results surrounding the work of the authors concerning boundary regularity of weak solutions of second order elliptic quasilinear equations in divergence form. The book also contains a complete development of regularity of solutions of variational inequalities, including the double obstacle problem, where the obstacles are allowed to be discontinuous. The book concludes with a chapter devoted to the existence theory thus providing the reader with a complete treatment of the subject ranging from regularity of weak solutions to the existence of weak solutions.

Author : Louis Dupaigne
Publisher : CRC Press
Page : 337 pages
File Size : 27,16 MB
Release : 2011-03-15
Category : Mathematics
ISBN : 1420066544

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Stable solutions are ubiquitous in differential equations. They represent meaningful solutions from a physical point of view and appear in many applications, including mathematical physics (combustion, phase transition theory) and geometry (minimal surfaces). Stable Solutions of Elliptic Partial Differential Equations offers a self-contained presentation of the notion of stability in elliptic partial differential equations (PDEs). The central questions of regularity and classification of stable solutions are treated at length. Specialists will find a summary of the most recent developments of the theory, such as nonlocal and higher-order equations. For beginners, the book walks you through the fine versions of the maximum principle, the standard regularity theory for linear elliptic equations, and the fundamental functional inequalities commonly used in this field. The text also includes two additional topics: the inverse-square potential and some background material on submanifolds of Euclidean space.

Author : Lucio Boccardo
Publisher : Walter de Gruyter
Page : 204 pages
File Size : 15,65 MB
Release : 2013-10-29
Category : Mathematics
ISBN : 3110315424

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Elliptic partial differential equations is one of the main and most active areas in mathematics. This book is devoted to the study of linear and nonlinear elliptic problems in divergence form, with the aim of providing classical results, as well as more recent developments about distributional solutions. For this reason this monograph is addressed to master's students, PhD students and anyone who wants to begin research in this mathematical field.

Author : Luigi Ambrosio
Publisher : Springer
Page : 230 pages
File Size : 33,32 MB
Release : 2019-01-10
Category : Mathematics
ISBN : 8876426515

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The book originates from the Elliptic PDE course given by the first author at the Scuola Normale Superiore in recent years. It covers the most classical aspects of the theory of Elliptic Partial Differential Equations and Calculus of Variations, including also more recent developments on partial regularity for systems and the theory of viscosity solutions.

Author : Lisa Beck
Publisher : Springer
Page : 214 pages
File Size : 36,32 MB
Release : 2016-04-08
Category : Mathematics
ISBN : 3319274856

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These lecture notes provide a self-contained introduction to regularity theory for elliptic equations and systems in divergence form. After a short review of some classical results on everywhere regularity for scalar-valued weak solutions, the presentation focuses on vector-valued weak solutions to a system of several coupled equations. In the vectorial case, weak solutions may have discontinuities and so are expected, in general, to be regular only outside of a set of measure zero. Several methods are presented concerning the proof of such partial regularity results, and optimal regularity is discussed. Finally, a short overview is given on the current state of the art concerning the size of the singular set on which discontinuities may occur. The notes are intended for graduate and postgraduate students with a solid background in functional analysis and some familiarity with partial differential equations; they will also be of interest to researchers working on related topics.

Author : Michel Chipot
Publisher : Springer Science & Business Media
Page : 289 pages
File Size : 50,81 MB
Release : 2009-02-19
Category : Mathematics
ISBN : 3764399813

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The aim of this book is to introduce the reader to different topics of the theory of elliptic partial differential equations by avoiding technicalities and refinements. Apart from the basic theory of equations in divergence form it includes subjects such as singular perturbation problems, homogenization, computations, asymptotic behaviour of problems in cylinders, elliptic systems, nonlinear problems, regularity theory, Navier-Stokes system, p-Laplace equation. Just a minimum on Sobolev spaces has been introduced, and work or integration on the boundary has been carefully avoided to keep the reader's attention on the beauty and variety of these issues. The chapters are relatively independent of each other and can be read or taught separately. Numerous results presented here are original and have not been published elsewhere. The book will be of interest to graduate students and faculty members specializing in partial differential equations.

Author : Luis Angel Caffarelli
Publisher : Edizioni della Normale
Page : 0 pages
File Size : 40,63 MB
Release : 1999-10-01
Category : Mathematics
ISBN : 9788876422492

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The material presented here corresponds to Fermi lectures that I was invited to deliver at the Scuola Normale di Pisa in the spring of 1998. The obstacle problem consists in studying the properties of minimizers of the Dirichlet integral in a domain D of Rn, among all those configurations u with prescribed boundary values and costrained to remain in D above a prescribed obstacle F. In the Hilbert space H1(D) of all those functions with square integrable gradient, we consider the closed convex set K of functions u with fixed boundary value and which are greater than F in D. There is a unique point in K minimizing the Dirichlet integral. That is called the solution to the obstacle problem.

Author : David Gilbarg
Publisher : Springer Science & Business Media
Page : 544 pages
File Size : 49,8 MB
Release : 2001-01-12
Category : Mathematics
ISBN : 9783540411604

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This work aims to be of interest to those who have to work with differential equations and acts either as a reference or as a book to learn from. The authors have made the treatment self-contained.

Author : Ariel Barton
Publisher : American Mathematical Soc.
Page : 120 pages
File Size : 30,14 MB
Release : 2013-04-22
Category : Mathematics
ISBN : 0821887408

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In this monograph the author investigates divergence-form elliptic partial differential equations in two-dimensional Lipschitz domains whose coefficient matrices have small (but possibly nonzero) imaginary parts and depend only on one of the two coordinates. He shows that for such operators, the Dirichlet problem with boundary data in $L^q$ can be solved for $q1$ small enough, and provide an endpoint result at $p=1$.