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BL
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| Record Number
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859686
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| Main Entry
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Edmunds, D. E., (David Eric)
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| Title & Author
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Elliptic differential operators and spectral analysis /\ David E. Edmunds, W. Desmond Evans.
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| Publication Statement
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Cham :: Springer,, 2018.
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| Series Statement
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Springer monographs in mathematics
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| Page. NO
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1 online resource
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| ISBN
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3030021254
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: 9783030021252
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3030021246
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9783030021245
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| Bibliographies/Indexes
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Includes bibliographical references and index.
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| Contents
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Intro; Preface; Contents; Basic Notation; 1 Preliminaries; 1.1 Integration; 1.2 Functional Analysis; 1.3 Function Spaces; 1.3.1 Spaces of Continuous Functions; 1.3.2 Sobolev Spaces; 1.4 The Hilbert and Riesz Transforms; 2 The Laplace Operator; 2.1 Mean Value Inequalities; 2.2 Representation of Solutions; 2.3 Dirichlet Problems: The Method of Perron; 2.4 Notes; 3 Second-Order Elliptic Equations; 3.1 Basic Notions; 3.2 Maximum Principles; 4 The Classical Dirichlet Problem for Second-Order Elliptic Operators; 4.1 Preamble; 4.2 The Poisson Equation; 4.3 More General Elliptic Operators; 4.4 Notes
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10.3 Results Involving the Laplace Operator10.4 The p-Laplacian; 11 More Properties of Sobolev Embeddings; 11.1 The Distance Function; 11.2 Nuclear Maps; 11.3 Asymptotic Formulae for Approximation Numbers of Sobolev Embeddings; 11.4 Spaces with Variable Exponent; 11.5 Notes; 12 The Dirac Operator; 12.1 Preamble; 12.2 The Dirac Equation; 12.3 The Free Dirac Operator; 12.4 The Brown-Ravenhall Operator; 12.5 Sums of Operators and Coulomb Potentials; 12.5.1 The Case A= mathbbD; 12.5.2 The Case A = mathbbH; 12.5.3 The Case A= mathbbB; 12.6 The Free Dirac Operator on a Bounded Domain
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5 Elliptic Operators of Arbitrary Order5.1 Preliminaries; 5.2 Gårding's Inequality; 5.3 The Dirichlet Problem; 5.4 A Little Regularity Theory; 5.5 Eigenvalues of the Laplacian; 5.6 Spectral Independence; 5.7 Notes; 6 Operators and Quadratic Forms in Hilbert Space; 6.1 Self-Adjoint Extensions of Symmetric Operators; 6.2 Characterisations of Self-Adjoint Extensions; 6.2.1 Linear Relations; 6.2.2 Boundary Triplets; 6.2.3 Gamma Fields and Weyl Functions; 6.3 The Friedrichs Extension; 6.4 The Krein-Vishik-Birman (KVB) Theory; 6.5 Adjoint Pairs and Closed Extensions; 6.6 Sectorial Operators
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7.3.3 Limit-Point and Limit-Circle Criteria7.4 Coercive Sectorial Operators; 7.4.1 The Case dim(kerT*) =2.; 7.5 Realisations of Second-Order Elliptic Operators on Domains; 7.6 Notes; 8 The Lp Approach to the Laplace Operator; 8.1 Preamble; 8.2 Technical Results; 8.3 Existence of a Weak Lp Solution; 8.4 Other Procedures; 8.5 Notes; 9 The p-Laplacian; 9.1 Preamble; 9.2 Preliminaries; 9.3 The Dirichlet Problem; 9.4 An Eigenvalue Problem; 9.5 More About the First Eigenvalue; 9.6 Notes; 10 The Rellich Inequality; 10.1 Preamble; 10.2 The Mean Distance Function
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| Abstract
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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.--
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| Subject
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Differential equations, Elliptic.
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| Subject
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Differential operators.
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Differential calculus equations.
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Differential equations, Elliptic.
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Differential operators.
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Functional analysis transforms.
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MATHEMATICS-- Calculus.
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MATHEMATICS-- Mathematical Analysis.
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Partial Differential Equations.
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Functional Analysis.
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Operator Theory.
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| Subject
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Ordinary Differential Equations.
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| Dewey Classification
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515.353
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| LC Classification
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QA377
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| Added Entry
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Evans, W. D.
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