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2018 | Buch

Preliminaries Kapitel lesen Erstes Kapitel lesen

Elliptic Differential Operators and Spectral Analysis

verfasst von: Prof. D. E. Edmunds, Prof. W.D. Evans

Verlag: Springer International Publishing

Buchreihe : Springer Monographs in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book deals with elliptic differential equations, providing the analytic background necessary for the treatment of associated spectral questions, and covering important topics previously scattered throughout the literature.

Starting with the basics of elliptic operators and their naturally associated function spaces, the authors then proceed to cover various related topics of current and continuing importance. Particular attention is given to the characterisation of self-adjoint extensions of symmetric operators acting in a Hilbert space and, for elliptic operators, the realisation of such extensions in terms of boundary conditions. A good deal of material not previously available in book form, such as the treatment of the Schauder estimates, is included.

Requiring only basic knowledge of measure theory and functional analysis, the book is accessible to graduate students and will be of interest to all researchers in partial differential equations. The reader will value its self-contained, thorough and unified presentation of the modern theory of elliptic operators.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Preliminaries
Abstract
This provides basic facts concerning integration, functional analysis, Sobolev spaces and the Hilbert and Riesz transforms.
David E. Edmunds, W. Desmond Evans
Chapter 2. The Laplace Operator
Abstract
The fundamental properties of harmonic and subharmonic functions are given together with maximum principles, the representation of solutions of the Poisson equation, Weyl’s lemma and Perron’s method for proving existence of solutions of the Dirichlet problem.
David E. Edmunds, W. Desmond Evans
Chapter 3. Second-Order Elliptic Equations
Abstract
Second-order elliptic equations are introduced and various forms of the maximum principle are provided.
David E. Edmunds, W. Desmond Evans
Chapter 4. The Classical Dirichlet Problem for Second-Order Elliptic Operators
Abstract
This studies the classical Dirichlet problem for second-order elliptic operators and presents familiar results such as Kellogg’s theorem and the Schauder boundary estimates. These estimates are established by use of the approach due to M. König that has not appeared in book form before.
David E. Edmunds, W. Desmond Evans
Chapter 5. Elliptic Operators of Arbitrary Order
Abstract
Various notions of ellipticity are given for operators of arbitrary order; Gårding’s inequality is proved and used to establish the existence of a weak solution of the Dirichlet problem. Returning to the Laplace operator, its Dirichlet eigenvalues are considered: Courant’s min-max pronciple is proved together with the analyticity of eigenvectors and the Faber-Krahn inequality for the first eigenvalue. There is a brief discussion of spectral independence.
David E. Edmunds, W. Desmond Evans
Chapter 6. Operators and Quadratic Forms in Hilbert Space
Abstract
This is devoted to self-adjoint extensions of symmetric opera- tors acting in a Hilbert space. The main purpose is to provide an account of the Krein-Vishik-Birman theory concerning the positive self-adjoint extensions of positive symmetric operators and the extension by Grubb of the theory to adjoint pairs of closed operators. Corresponding results, due to Arlinski and his collaborators, concerning the m-sectorial extensions of sectorial operators are given.
David E. Edmunds, W. Desmond Evans
Chapter 7. Realisations of Second-Order Linear Elliptic Operators
Abstract
Here the abstract results of the previous chapter are applied to give realisations of second-order elliptic operators. The first four sections deal with symmetric Sturm–Liouville operators and lead to a description of coercive sectorial operators of this type. An outline is given of the work of Grubb in which her abstract theory is applied to uniformly elliptic operators generated by differential expressions A, leading to the identification of all closed realisations of A by means of boundary conditions expressed in terms of differential operators acting between function spaces defined on the boundary.
David E. Edmunds, W. Desmond Evans
Chapter 8. The Approach to the Laplace Operator
Abstract
The Poisson equation \(\Delta u=f\) on a bounded open subset of \(\mathbb {R}^n\) is considered when f belongs to \(L_p(\Omega )\) for some p between 1 and 2 but does not belong to \(L_2(\Omega )\): The Hilbert space methods of earlier chapters are then not applicable, but use of a technique due to Simader and Sohr is shown to give a type of weak solution in the context of \(L_p\).
David E. Edmunds, W. Desmond Evans
Chapter 9. The Laplacian
Abstract
This provides a small selection of the enormous amount of information now available for the p–Laplacian. The existence of a solution of the Dirichlet problem is proved, together with a number of results concerning eigenvalues, including a version of the Courant nodal domain theorem.
David E. Edmunds, W. Desmond Evans
Chapter 10. The Rellich Inequality
Abstract
Here the Rellich inequality is studied in the context of \(L_p(\Omega )\), where \(\Omega \) is an open subset \(\Omega \) of \(\mathbb {R}^n\). The classical situation is that in which \(p = 2\) and \(\Omega = \mathbb {R}^{n}\backslash \{0\} \); here analogous results in the general case are obtained by use of a mean distance function.
David E. Edmunds, W. Desmond Evans
Chapter 11. More Properties of Sobolev Embeddings
Abstract
Various properties of the Sobolev embeddings that feature in the study of partial differential equations are given. These include necessary and suffcient conditions for a Sobolev embedding to be nuclear, and a characterisation of the subspace of a Sobolev space consisting of functions with zero trace.
David E. Edmunds, W. Desmond Evans
Chapter 12. The Dirac Operator
Abstract
The concern here is with positive operators which model relativistic properties of the Dirac operator, special attention being given to the Brown-Ravenhall operator defined on a domain \(\Omega \) properly contained in \(\mathbb {R}^n\).
David E. Edmunds, W. Desmond Evans
Backmatter
Metadaten
Titel
Elliptic Differential Operators and Spectral Analysis
verfasst von
Prof. D. E. Edmunds
Prof. W.D. Evans
Copyright-Jahr
2018
Electronic ISBN
978-3-030-02125-2
Print ISBN
978-3-030-02124-5
DOI
https://doi.org/10.1007/978-3-030-02125-2