The Dirichlet Problem for Relative Harmonic Functions

Abstract

AbstractThe class of relative harmonic functions is suggested by the following trivial remark. Let (D, OC (M, p)D) be a measurable space, and suppose that to each point ξ of D is assigned some set (perhaps empty) {μα (ξ, ·), α ∈ I ξ} of probability measures on D. Call a function generalized harmonic if it satisfies specified smoothness conditions and if $$ v(\zeta ) = \int_D {v(\eta ({\mu_{\alpha }}(} \zeta, d\eta ) = {\mu_{\alpha }}(\zeta, v) $$ (1.1) for ξ in D and α in I ξ. For example, if D is an open subset of ℝN, if for each ξ the index α represents a ball B of center ξ with closure in D, if I ξ is the class of all such balls, and if μB(ξ, v) is the unweighted average of v on ∂B, then the class of continuous functions on D satisfying (1.1) is the class of harmonic functions on D. Going back to the general case, suppose that h is a strictly positive generalized harmonic function and define μh α(ξ, ·) by $$ \mu_{\alpha }^h(\zeta, A) = \int_A {h(\eta )\frac{{{\mu_{\alpha }}(\zeta, d\eta )}}{{h(\zeta )}}} $$ (1.2) KeywordsHarmonic FunctionBoundary PointDirichlet ProblemBoundary FunctionHarmonic MeasureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.