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Direct Methods in the Theory of Elliptic Equations / by Jindřich Nečas
Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Springer Monographs in Mathematics | SpringerLink BücherPublisher: Berlin, Heidelberg : Springer-Verlag Berlin Heidelberg, 2012Description: Online-Ressource (XVI, 372p. 11 illus, digital)ISBN:- 9783642104558
- 1283369621
- 9781283369626
- 515/.3533 23
- 515.353
- 510
- QA370-380
- QA377
- Online Book
Contents:
Summary: 1.Introduction to the problem -- 2.Sobolev spaces -- 3.Exitence, Uniqueness of basic problems -- 4.Regularity of solution -- 5.Applications of Rellich’s inequalities and generalization to boundary value problems -- 6.Sobolev spaces with weights and applications to the boundary value problems -- 7.Regularity of solutions in case of irregular domains and elliptic problems with variable coefficients.Summary: 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.PPN: PPN: 1651089116Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
Direct Methods in the Theory of Elliptic Equations; Preface; About the Translation; Preface to the French Edition; Contents; Chapter 1 Elementary Description of Principal Results; 1.1 Beppo Levi Spaces; 1.1.1 Definition of Wk,2; 1.1.2 Equivalent Norms; 1.1.3 Concept of a Trace; 1.1.4 The Poincaré Inequality; 1.1.5 Rellich's Theorem; 1.1.6 The Generalized Poincaré Inequality; 1.1.7 The Quotient Spaces; 1.1.8 Other Equivalent Norms; 1.1.9 An Imbedding Theorem; 1.2 Boundary Value Problems for Elliptic Operators; 1.2.1 Elliptic Operators; 1.2.2 Decomposition of Operators
1.2.3 The Boundary Operators1.2.4 Green's Formula; 1.2.5 Sesquilinear Forms; 1.2.6 Boundary Value Problems; 1.2.7 Examples; 1.3 The V-ellipticity, Existence and Uniqueness of the Solution; 1.3.1 The Lax-Milgram Lemma; 1.3.2 Solving the Boundary Value Problem; 1.3.3 The Case of Quotient Spaces; 1.3.4 Conditions of V-ellipticity; 1.3.5 Nonstable Boundary Conditions; 1.3.6 Orthogonal Projections; 1.4 The Ritz, Galerkin, and Least Squares Methods; 1.4.1 The Variational Method; 1.4.2 The Galerkin Method; 1.4.3 The Least Squares Method; 1.5 Basic Notions from the Spectral Theory
1.5.1 Eigenvalues and Eigenfunctions, the Fredholm Alternative1.5.2 Eigenvalues and Eigenfunctions, the Fredholm Alternative (Continuation); 1.5.3 The Gårding Inequality; Chapter 2 The Spaces Wk,p; 2.1 Definitions and Auxiliary Theorems; 2.1.1 Classification of Domains, Pseudotopology in C80(O); 2.1.2 The Space Lp(O), Mean Continuity; 2.1.3 The Regularizing Operator; 2.1.4 Compactness Condition; 2.2 The Spaces Wk,p(O); 2.2.1 A Property of the Regularizing Operator; 2.2.2 The Absolute Continuity; 2.2.3 The Spaces Wk,p(O); 2.2.4 The Spaces Wk,p(O) (Continuation); 2.2.5 The Spaces Wk,p0(O)
2.3 Imbedding Theorems2.3.1 The Lipschitz Transform; 2.3.2 Density of C8(O) in Wk,p(O); 2.3.3 The Gagliardo Lemma; 2.3.4 The Sobolev Imbedding Theorems; 2.3.5 The Sobolev Imbedding Theorems (Continuation); 2.3.6 Extension, the Nikolskii Method; 2.3.7 Extension, the Calderon Method; 2.3.8 The Spaces Wk,p(O), k Non-integer; 2.4 The Problem of Traces; 2.4.1 Lemmas; 2.4.2 Imbedding Theorems; 2.4.3 Two Trace Theorems; 2.4.4 Some Other Properties of Traces; 2.5 The Problem of Traces (Continuation); 2.5.1 Application of the Fourier Transform; 2.5.2 Lemmas Based on the Hardy Inequality
2.5.3 Imbedding Theorems, Application of the Spaces W1-1/p,p(dO)2.5.4 Imbedding Theorems, Application of the Spaces W1-1/p,p(dO) (Continuation); 2.5.5 A Lemma; 2.5.6 The Converse Theorem; 2.5.7 The Converse Theorem (Continuation); 2.5.8 Remarks; 2.6 Compactness; 2.6.1 The Kondrashov Theorem; 2.6.2 Traces; 2.6.3 The Lions Lemma, Another Theorem of Compactness; 2.7 Quotient Spaces, Equivalent Norms; 2.7.1 Equivalent Norms; 2.7.2 Quotient Spaces; 2.7.3 The Spaces Vk,p(O); 2.7.4 Nikodym Domains; Chapter 3 Existence, Uniqueness and Fundamental Properties of Solutions of Boundary Value Problems
3.1 The Boundary Integral, Green's Formula
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