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

Given a compact doubling metric measure space X that supports a 2-Poincaré inequality, we construct a Dirichlet form on N1,2(X) N 1 , 2 ( X ) that is comparable to the upper gradient energy form on N1,2(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 N1,2(X) N 1 , 2 ( X ) using the Dirichlet form on the graph. We show that the Γ 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 Ω ⊆ 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.