Classical Potential TheorySpringer Science & Business Media, 6 de dez. de 2012 - 333 páginas 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. |
Conteúdo
| 8 | |
Subharmonic Functions | 59 |
4 | 89 |
Polar Sets and Capacity | 123 |
The Dirichlet Problem | 163 |
The Fine Topology | 197 |
Boundary Limits | 273 |
Appendix | 305 |
| 317 | |
| 329 | |
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Termos e frases comuns
ball Borel boundary point bounded open set choose compact set compact subset component continuous convex Corollary countable define Definition denote Dirichlet problem dµ(y E S(N Exercise exists finite follows from Theorem function h greatest harmonic minorant Green function H(RN h₁ h₂ harmonic functions harmonic minorant harmonic polynomial Hence holds holomorphic increasing sequence inequality Jy,m Kelvin transform Lemma Let f let ƒ Let h lim sup locally uniformly lower semicontinuous maximum principle mean value property minimal fine limit minimally thin minorant of Uy neighbourhood non-negative non-thin obtain open set open subset Poisson integral polar set positive number potential Proof prove quasi-everywhere r₁ regular result RN\K satisfies subharmonic functions subset of RN superharmonic function suppose t₁ topology U+(N u₁ unique upper semicontinuous v₁ w₁