# The Dirichlet Problem for Relative Harmonic Functions

Chapter

## Abstract

The class of relative harmonic functions is suggested by the following trivial remark. Let (
for ξ in

*D*, OC (M, p)D) be a measurable space, and suppose that to each point ξ of*D*is assigned some set (perhaps empty) {μ_{α}(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)

*D*and α in*I*_{ξ}. For example, if*D*is an open subset of ℝ^{ N }, 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 μ_{B}(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 μ^{h}_{α}(ξ, ·) by$$ \mu_{\alpha }^h(\zeta, A) = \int_A {h(\eta )\frac{{{\mu_{\alpha }}(\zeta, d\eta )}}{{h(\zeta )}}} $$

(1.2)

## Keywords

Harmonic Function Boundary Point Dirichlet Problem Boundary Function Harmonic Measure
These 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.

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© Springer-Verlag Berlin Heidelberg 2001