The Dirichlet Problem for Relative Harmonic Functions

The 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) {μ_α (ξ, ·), α ∈ I_ξ} of probability measures on D. Call a function generalized harmonic if it satisfies specified smoothness conditions and if 1.1$$ v(\zeta ) = \int_D {v(\eta ({\mu_{\alpha }}(} \zeta, d\eta ) = {\mu_{\alpha }}(\zeta, v) $$ 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 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 μh_α(ξ, ·) by 1.2$$ \mu_{\alpha }^h(\zeta, A) = \int_A {h(\eta )\frac{{{\mu_{\alpha }}(\zeta, d\eta )}}{{h(\zeta )}}} $$