Author | Armitage, David H. author |
---|---|
Title | Classical Potential Theory [electronic resource] / by David H. Armitage, Stephen J. Gardiner |
Imprint | London : Springer London : Imprint: Springer, 2001 |
Connect to | http://dx.doi.org/10.1007/978-1-4471-0233-5 |
Descript | XVI, 333 p. online resource |
1. Harmonic Functions -- 1.1. Laplaceโs equation -- 1.2. The mean value property -- 1.3. The Poisson integral for a ball -- 1.4. Harnackโs 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. Wienerโ 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-Naรฏm-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