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.

翻译：我们研究了在有界域上、受传统局部狄利克雷边界条件约束的一类非局部扩散模型所关联的非局部狄利克雷能量。本文的目标是提供一个通用框架，以在非局部扩散模型中正确施加狄利克雷边界条件。为此，我们将狄利克雷边界条件表述为一个惩罚项，并利用Γ收敛理论来研究惩罚项的正确形式。基于对Γ收敛的分析，我们证明：只要惩罚项满足若干温和条件，狄利克雷边界条件就能在Γ收敛的意义下于非局部扩散模型中正确施加。这项工作为非局部扩散模型中近似狄利克雷边界条件提供了理论基础。