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 \(N^{1,2}(X)\) $N^{1,2}(X)$ 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 \(\Gamma \) $\mathrm{\Gamma }$ -limit \(\mathcal {E}\) $\mathcal{E}$ of this family of bilinear forms (by taking a subsequence) exists and that \(\mathcal {E}\) $\mathcal{E}$ is a Dirichlet form on X. Properties of \(\mathcal {E}\) $\mathcal{E}$ are established. Moreover, we prove that \(\mathcal {E}\) $\mathcal{E}$ has the property of matching boundary values on a domain \(\Omega \subseteq X\) $\mathrm{\Omega }\subseteq X$ . This construction makes it possible to approximate harmonic functions (with respect to the Dirichlet form \(\mathcal {E}\) $\mathcal{E}$ ) 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.