Construction of a Dirichlet form on Metric Measure Spaces of Controlled Geometry

Abstract

Given a compact doubling metric measure space X that supports a 2-Poincaré inequality, we construct a Dirichlet form on $$N^{1,2}(X)$$ N 1 , 2 ( X ) that is comparable to the upper gradient energy form on $$N^{1,2}(X)$$ N 1 , 2 ( X ) . Our approach is based on the approximation of X by a family of graphs that is doubling and supports a 2-Poincaré inequality (see [20]). We construct a bilinear form on $$N^{1,2}(X)$$ N 1 , 2 ( X ) using the Dirichlet form on the graph. We show that the $$\Gamma $$ Γ -limit $$\mathcal {E}$$ E of this family of bilinear forms (by taking a subsequence) exists and that $$\mathcal {E}$$ E is a Dirichlet form on X. Properties of $$\mathcal {E}$$ E are established. Moreover, we prove that $$\mathcal {E}$$ E has the property of matching boundary values on a domain $$\Omega \subseteq X$$ Ω ⊆ X . This construction makes it possible to approximate harmonic functions (with respect to the Dirichlet form $$\mathcal {E}$$ E ) on a domain in X with a prescribed Lipschitz boundary data via a numerical scheme dictated by the approximating Dirichlet forms, which are discrete objects.