Classical Potential Theory [Hardcover]

A long-awaited, updated introductory text by the world leaders in potential theory. This essential reference work covers all aspects of this major field of mathematical research, from basic theory and exercises to more advanced topological ideas. The largely self-contained presentation makes it basically accessible to graduate students.

From its origins in Newtonian physics, potential theory has developed into a major field of mathematical research. This book provides a comprehensive treatment of classical potential theory: it covers harmonic and subharmonic functions, maximum principles, polynomial expansions, Green functions, potentials and capacity, the Dirichlet problem and boundary integral representations. The first six chapters deal concretely with the basic theory, and include exercises. The final three chapters are more advanced and treat topological ideas specifically created for potential theory, such as the fine topology, the Martin boundary and minimal thinness.
The presentation is largely self-contained and is accessible to graduate students, the only prerequisites being a reasonable grounding in analysis and several variables calculus, and a first course in measure theory. The book will prove an essential reference to all those with an interest in potential theory and its applications.1. Harmonic Functions.- 1.1. Laplaces equation.- 1.2. The mean value property.- 1.3. The Poisson integral for a ball.- 1.4. Harnacks 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. HarmonlÃ