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 Γ -limit E E of this family of bilinear forms (by taking a subsequence) exists and that E E is a Dirichlet form on X. Properties of E E are established. Moreover, we prove that E E has the property of matching boundary values on a domain X Ω ⊆ X. This construction makes it possible to approximate harmonic functions (with respect to the Dirichlet form 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.