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.
Elliptic Regularity Theory
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.
Regularity Theory for Quasilinear Elliptic Systems and Monge - Ampere Equations in Two Dimensions
Regularity Theory for Quasilinear Elliptic Systems and Monge - Ampere Equations in Two Dimensions - 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 Theory for Quasilinear Elliptic Systems and Monge - Ampere Equations in Two Dimensions write by Friedmar Schulz. This book was released on 2006-12-08. Regularity Theory for Quasilinear Elliptic Systems and Monge - Ampere Equations in Two Dimensions available in PDF, EPUB and Kindle. These lecture notes have been written as an introduction to the characteristic theory for two-dimensional Monge-Ampère equations, a theory largely developed by H. Lewy and E. Heinz which has never been presented in book form. An exposition of the Heinz-Lewy theory requires auxiliary material which can be found in various monographs, but which is presented here, in part because the focus is different, and also because these notes have an introductory character. Self-contained introductions to the regularity theory of elliptic systems, the theory of pseudoanalytic functions and the theory of conformal mappings are included. These notes grew out of a seminar given at the University of Kentucky in the fall of 1988 and are intended for graduate students and researchers interested in this area.
Periodic Homogenization of Elliptic Systems
Periodic Homogenization of Elliptic Systems - 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 Periodic Homogenization of Elliptic Systems write by Zhongwei Shen. This book was released on 2018-09-04. Periodic Homogenization of Elliptic Systems available in PDF, EPUB and Kindle. This monograph surveys the theory of quantitative homogenization for second-order linear elliptic systems in divergence form with rapidly oscillating periodic coefficients in a bounded domain. It begins with a review of the classical qualitative homogenization theory, and addresses the problem of convergence rates of solutions. The main body of the monograph investigates various interior and boundary regularity estimates that are uniform in the small parameter e>0. Additional topics include convergence rates for Dirichlet eigenvalues and asymptotic expansions of fundamental solutions, Green functions, and Neumann functions. The monograph is intended for advanced graduate students and researchers in the general areas of analysis and partial differential equations. It provides the reader with a clear and concise exposition of an important and currently active area of quantitative homogenization.
Regularity of Minimal Surfaces
Regularity of Minimal Surfaces - 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 of Minimal Surfaces write by Ulrich Dierkes. This book was released on 2010-08-16. Regularity of Minimal Surfaces available in PDF, EPUB and Kindle. Regularity of Minimal Surfaces begins with a survey of minimal surfaces with free boundaries. Following this, the basic results concerning the boundary behaviour of minimal surfaces and H-surfaces with fixed or free boundaries are studied. In particular, the asymptotic expansions at interior and boundary branch points are derived, leading to general Gauss-Bonnet formulas. Furthermore, gradient estimates and asymptotic expansions for minimal surfaces with only piecewise smooth boundaries are obtained. One of the main features of free boundary value problems for minimal surfaces is that, for principal reasons, it is impossible to derive a priori estimates. Therefore regularity proofs for non-minimizers have to be based on indirect reasoning using monotonicity formulas. This is followed by a long chapter discussing geometric properties of minimal and H-surfaces such as enclosure theorems and isoperimetric inequalities, leading to the discussion of obstacle problems and of Plateau ́s problem for H-surfaces in a Riemannian manifold. A natural generalization of the isoperimetric problem is the so-called thread problem, dealing with minimal surfaces whose boundary consists of a fixed arc of given length. Existence and regularity of solutions are discussed. The final chapter on branch points presents a new approach to the theorem that area minimizing solutions of Plateau ́s problem have no interior branch points.
