Classical Potential Theory
Springer Science & Business Media, 2001 - Mathematics - 333 pages
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.
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arbitrary Borel bounded open set Brelot choose compact set compact subset component contains convex Corollary countable define Definition denote Dirichlet problem domains Exercise exists fine topology finite follows from Theorem function f greatest harmonic minorant Green function harmonic functions harmonic polynomial Harnack Harnack's inequalities Hence holds increasing sequence irregular boundary point Kelvin transform Lemma let f Let h liminf limit point limsup locally uniformly lower semicontinuous Martin boundary Martin topology 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 constant positive harmonic function positive number potential Proof prove quasi.everywhere regular respect Riesz measure associated RN\K satisfies subharmonic functions subset of Q superharmonic function suppose U+(Q unique upper semicontinuous Uy(x
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