Approximation of harmonic functions on metric measure spaces of controlled geometry via discrete graphs

TL;DR

This paper develops a method to approximate harmonic functions on certain metric measure spaces using discrete graphs, establishing convergence and energy minimization properties.

Contribution

It introduces a novel graph-based approximation scheme for harmonic functions on doubling metric measure spaces supporting a Poincaré inequality.

Findings

Harmonic functions are approximated as weak limits of graph minimizers.

The energy form on the domain is obtained as a Gamma-limit of discrete energy forms.

The approach applies to spaces with controlled geometry and boundary data.

Abstract

Given a complete doubling metric measure space $X$ that supports a $2$-Poincar\'e inequality, we approximate harmonic functions on a bounded domain $\Omega$ with a prescribed Newton-Sobolev boundary data. Our approach is based on the approximation of the underlying space $X$ by a family of graphs. This approximated harmonic function is realized as the weak limit of a sequence of functions obtained from the graph minimizers. We prove that such a function is a minimizer with respect to a nonlinear energy form on $N^{1,2}_0(\Omega)$, which is in turn, majorized by the upper gradient energy on $N^{1,2}(X)$. This energy form on $N^{1,2}_0(\Omega)$ is obtained as a $\Gamma$-limit of a sequence of induced energy forms projected from the discrete energy form on the approximating graphs.