$\Gamma$-convergence of Nonlocal Dirichlet Energies With Penalty Formulations of Dirichlet Boundary Data

Abstract

We study nonlocal Dirichlet energies associated with a class of nonlocal diffusion models on a bounded domain subject to the conventional local Dirichlet boundary condition. The goal of this paper is to give a general framework to correctly impose Dirichlet boundary condition in nonlocal diffusion model. To achieve this, we formulate the Dirichlet boundary condition as a penalty term and use theory of $\Gamma$-convergence to study the correct form of the penalty term. Based on the analysis of $\Gamma$-convergence, we prove that the Dirichlet boundary condition can be correctly imposed in nonlocal diffusion model in the sense of $\Gamma$-convergence as long as the penalty term satisfies a few mild conditions. This work provides a theoretical foundation for approximate Dirichlet boundary condition in nonlocal diffusion model.