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

TL;DR

This paper establishes a theoretical framework using $ ext{Γ}$-convergence to correctly impose Dirichlet boundary conditions in nonlocal diffusion models via penalty formulations, ensuring proper boundary behavior in the limit.

Contribution

It introduces a general $ ext{Γ}$-convergence approach to formulate and analyze penalty terms for Dirichlet boundary conditions in nonlocal diffusion models, providing a rigorous foundation.

Findings

$ ext{Γ}$-convergence characterizes the limit of penalty-based nonlocal energies.

Properly chosen penalty terms ensure correct boundary conditions in the limit.

The framework applies to a broad class of nonlocal diffusion models.

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 $\varGamma$-convergence to study the correct form of the penalty term. Based on the analysis of $\varGamma$-convergence, we prove that the Dirichlet boundary condition can be correctly imposed in nonlocal diffusion model in the sense of $\varGamma$-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.