Beschreibung
Inhaltsangabe1. Harmonic Functions.- 1.1. Laplace's equation.- 1.2. The mean value property.- 1.3. The Poisson integral for a ball.- 1.4. Harnack's inequalities.- 1.5. Families of harmonic functions: convergence properties.- 1.6. The Kelvin transform.- 1.7. Harmonic functions on half-spaces.- 1.8. Real-analyticity of harmonic functions.- 1.9. Exercises.- 2. Harmonic Polynomials.- 2.1. Spaces of homogeneous polynomials.- 2.2. Another inner product on a space of polynomials.- 2.3. Axially symmetric harmonic polynomials.- 2.4. Polynomial expansions of harmonic functions.- 2.5. Laurent expansions of harmonic functions.- 2.6. Harmonic approximation.- 2.7. Harmonic polynomials and classical polynomials.- 2.8. Exercises.- 3. Subharmonic Functions.- 3.1. Elementary properties.- 3.2. Criteria for subharmonicity.- 3.3. Approximation of subharmonic functions by smooth ones.- 3.4. Convexity and subharmonicity.- 3.5. Mean values and subharmonicity.- 3.6. Harmonic majorants.- 3.7. Families of subharmonic functions: convergence properties.- 3.8. Exercises.- 4. Potentials.- 4.1. Green functions.- 4.2. Potentials.- 4.3. The distributional Laplacian.- 4.4. The Riesz decomposition.- 4.5. Continuity and smoothness properties.- 4.6. Classical boundary limit theorems.- 4.7. Exercises.- 5. Polar Sets and Capacity.- 5.1. Polar sets.- 5.2. Removable singularity theorems.- 5.3. Reduced functions.- 5.4. The capacity of a compact set.- 5.5. Inner and outer capacity.- 5.6. Capacitable sets.- 5.7. The fundamental convergence theorem.- 5.8. Logarithmic capacity.- 5.9. Hausdorff measure and capacity.- 5.10. Exercises.- 6. The Dirichlet Problem.- 6.1. Introduction.- 6.2. Upper and lower PWB solutions.- 6.3. Further properties of PWB solutions.- 6.4. Harmonic measure.- 6.5. Negligible sets.- 6.6. Boundary behaviour.- 6.7. Behaviour near infinity.- 6.8. Regularity and the Green function.- 6.9. PWB solutions and reduced functions.- 6.10. Superharmonic extension.- 6.11. Exercises.- 7. The Fine Topology.- 7.1. Introduction.- 7.2. Thin sets.- 7.3. Thin sets and reduced functions.- 7.4. Fine limits.- 7.5. Thin set s and the Dirichlet problem.- 7.6. Thinness at infinity.- 7.7. Wiener' s criterion.- 7.8. Limit properties of superharmonic functions.- 7.9. Harmonic approximation.- 8. The Martin Boundary.- 8.1. The Martin kernel and Mart in boundary.- 8.2. Reduced functions and minimal harmonic functions.- 8.3. Reduction ?0s and ?1.- 8.4. The Martin representation.- 8.5. The Martin boundary of a strip.- 8.6. The Martin kernel and the Kelvin transform.- 8.7. The boundary Harnack principle for Lipschitz domains.- 8.8. The Marti n boundary of a Lipschitz domain.- 9. Boundary Limits.- 9.1. Swept measures and the Dirichlet problem for the Martin compactification.- 9.2. Minimal thinness.- 9.3. Minimal fine limits.- 9.4. The Fatou-Naïm-Doob theorem.- 9.5. Minimal thinness in subdomains.- 9.6. Refinements of limit theorems.- 9.7. Minimal thinness in a half-space.- Historical Notes.- References.- Symbol Index.
Autorenporträt
Inhaltsangabe1. Harmonic Functions.- 1.1. Laplace's equation.- 1.2. The mean value property.- 1.3. The Poisson integral for a ball.- 1.4. Harnack's inequalities.- 1.5. Families of harmonic functions: convergence properties.- 1.6. The Kelvin transform.- 1.7. Harmonic functions on half-spaces.- 1.8. Real-analyticity of harmonic functions.- 1.9. Exercises.- 2. Harmonic Polynomials.- 2.1. Spaces of homogeneous polynomials.- 2.2. Another inner product on a space of polynomials.- 2.3. Axially symmetric harmonic polynomials.- 2.4. Polynomial expansions of harmonic functions.- 2.5. Laurent expansions of harmonic functions.- 2.6. Harmonic approximation.- 2.7. Harmonic polynomials and classical polynomials.- 2.8. Exercises.- 3. Subharmonic Functions.- 3.1. Elementary properties.- 3.2. Criteria for subharmonicity.- 3.3. Approximation of subharmonic functions by smooth ones.- 3.4. Convexity and subharmonicity.- 3.5. Mean values and subharmonicity.- 3.6. Harmonic majorants.- 3.7. Families of subharmonic functions: convergence properties.- 3.8. Exercises.- 4. Potentials.- 4.1. Green functions.- 4.2. Potentials.- 4.3. The distributional Laplacian.- 4.4. The Riesz decomposition.- 4.5. Continuity and smoothness properties.- 4.6. Classical boundary limit theorems.- 4.7. Exercises.- 5. Polar Sets and Capacity.- 5.1. Polar sets.- 5.2. Removable singularity theorems.- 5.3. Reduced functions.- 5.4. The capacity of a compact set.- 5.5. Inner and outer capacity.- 5.6. Capacitable sets.- 5.7. The fundamental convergence theorem.- 5.8. Logarithmic capacity.- 5.9. Hausdorff measure and capacity.- 5.10. Exercises.- 6. The Dirichlet Problem.- 6.1. Introduction.- 6.2. Upper and lower PWB solutions.- 6.3. Further properties of PWB solutions.- 6.4. Harmonic measure.- 6.5. Negligible sets.- 6.6. Boundary behaviour.- 6.7. Behaviour near infinity.- 6.8. Regularity and the Green function.- 6.9. PWB solutions and reduced functions.- 6.10. Superharmonic extension.- 6.11. Exercises.- 7. The Fine Topology.- 7.1. Introduction.- 7.2. Thin sets.- 7.3. Thin sets and reduced functions.- 7.4. Fine limits.- 7.5. Thin set s and the Dirichlet problem.- 7.6. Thinness at infinity.- 7.7. Wiener' s criterion.- 7.8. Limit properties of superharmonic functions.- 7.9. Harmonic approximation.- 8. The Martin Boundary.- 8.1. The Martin kernel and Mart in boundary.- 8.2. Reduced functions and minimal harmonic functions.- 8.3. Reduction ?0s and ?1.- 8.4. The Martin representation.- 8.5. The Martin boundary of a strip.- 8.6. The Martin kernel and the Kelvin transform.- 8.7. The boundary Harnack principle for Lipschitz domains.- 8.8. The Marti n boundary of a Lipschitz domain.- 9. Boundary Limits.- 9.1. Swept measures and the Dirichlet problem for the Martin compactification.- 9.2. Minimal thinness.- 9.3. Minimal fine limits.- 9.4. The Fatou-Naïm-Doob theorem.- 9.5. Minimal thinness in subdomains.- 9.6. Refinements of limit theorems.- 9.7. Minimal thinness in a half-space.- Historical Notes.- References.- Symbol Index.
