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Table of Contents
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
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
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
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
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
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
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