Classical Potential Theory

1. 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. Boundarybehaviour. - 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. Language: English