Elliptic Regularity Theory - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Elliptic Regularity Theory write by Lisa Beck. This book was released on 2016-04-08. Elliptic Regularity Theory available in PDF, EPUB and Kindle. 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.
An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs
An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs write by Mariano Giaquinta. This book was released on 2013-07-30. An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs available in PDF, EPUB and Kindle. This volume deals with the regularity theory for elliptic systems. We may find the origin of such a theory in two of the problems posed by David Hilbert in his celebrated lecture delivered during the International Congress of Mathematicians in 1900 in Paris: 19th problem: Are the solutions to regular problems in the Calculus of Variations always necessarily analytic? 20th problem: does any variational problem have a solution, provided that certain assumptions regarding the given boundary conditions are satisfied, and provided that the notion of a solution is suitably extended? During the last century these two problems have generated a great deal of work, usually referred to as regularity theory, which makes this topic quite relevant in many fields and still very active for research. However, the purpose of this volume, addressed mainly to students, is much more limited. We aim to illustrate only some of the basic ideas and techniques introduced in this context, confining ourselves to important but simple situations and refraining from completeness. In fact some relevant topics are omitted. Topics include: harmonic functions, direct methods, Hilbert space methods and Sobolev spaces, energy estimates, Schauder and L^p-theory both with and without potential theory, including the Calderon-Zygmund theorem, Harnack's and De Giorgi-Moser-Nash theorems in the scalar case and partial regularity theorems in the vector valued case; energy minimizing harmonic maps and minimal graphs in codimension 1 and greater than 1. In this second deeply revised edition we also included the regularity of 2-dimensional weakly harmonic maps, the partial regularity of stationary harmonic maps, and their connections with the case p=1 of the L^p theory, including the celebrated results of Wente and of Coifman-Lions-Meyer-Semmes.
Fine Regularity of Solutions of Elliptic Partial Differential Equations
Fine Regularity of Solutions of Elliptic Partial Differential Equations - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Fine Regularity of Solutions of Elliptic Partial Differential Equations write by Jan Malý. This book was released on 1997. Fine Regularity of Solutions of Elliptic Partial Differential Equations available in PDF, EPUB and Kindle. 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.
Direct Methods in the Theory of Elliptic Equations
Direct Methods in the Theory of Elliptic Equations - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Direct Methods in the Theory of Elliptic Equations write by Jindrich Necas. This book was released on 2011-10-06. Direct Methods in the Theory of Elliptic Equations available in PDF, EPUB and Kindle. Nečas’ book Direct Methods in the Theory of Elliptic Equations, published 1967 in French, has become a standard reference for the mathematical theory of linear elliptic equations and systems. This English edition, translated by G. Tronel and A. Kufner, presents Nečas’ work essentially in the form it was published in 1967. It gives a timeless and in some sense definitive treatment of a number issues in variational methods for elliptic systems and higher order equations. The text is recommended to graduate students of partial differential equations, postdoctoral associates in Analysis, and scientists working with linear elliptic systems. In fact, any researcher using the theory of elliptic systems will benefit from having the book in his library. The volume gives a self-contained presentation of the elliptic theory based on the "direct method", also known as the variational method. Due to its universality and close connections to numerical approximations, the variational method has become one of the most important approaches to the elliptic theory. The method does not rely on the maximum principle or other special properties of the scalar second order elliptic equations, and it is ideally suited for handling systems of equations of arbitrary order. The prototypical examples of equations covered by the theory are, in addition to the standard Laplace equation, Lame’s system of linear elasticity and the biharmonic equation (both with variable coefficients, of course). General ellipticity conditions are discussed and most of the natural boundary condition is covered. The necessary foundations of the function space theory are explained along the way, in an arguably optimal manner. The standard boundary regularity requirement on the domains is the Lipschitz continuity of the boundary, which "when going beyond the scalar equations of second order" turns out to be a very natural class. These choices reflect the author's opinion that the Lame system and the biharmonic equations are just as important as the Laplace equation, and that the class of the domains with the Lipschitz continuous boundary (as opposed to smooth domains) is the most natural class of domains to consider in connection with these equations and their applications.
Regularity Results for Nonlinear Elliptic Systems and Applications
Regularity Results for Nonlinear Elliptic Systems and Applications - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Regularity Results for Nonlinear Elliptic Systems and Applications write by Alain Bensoussan. This book was released on 2013-04-17. Regularity Results for Nonlinear Elliptic Systems and Applications available in PDF, EPUB and Kindle. This book collects many helpful techniques for obtaining regularity results for solutions of nonlinear systems of partial differential equations. These are applied in various cases to provide useful examples and relevant results, particularly in such fields as fluid mechanics, solid mechanics, semiconductor theory and game theory.
