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

Weiye Gan, Qiang Du, Zuoqiang Shi

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.

$Γ$-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 -convergence to study the correct form of the penalty term. Based on the analysis of -convergence, we prove that the Dirichlet boundary condition can be correctly imposed in nonlocal diffusion model in the sense of -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.
Paper Structure (12 sections, 19 theorems, 121 equations)

This paper contains 12 sections, 19 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Assumptions and main results
  3. Quantitative discussion
  4. Preliminaries
  5. --Convergence
  6. Relation between kernels of different scales
  7. --Convergence of nonlocal functionals
  8. Nonlocal model for Dirichlet problem
  9. Nonlocal model for eigenvalue problem
  10. Nonlocal model with general boudary term
  11. Compactness
  12. Conclusion

Key Result

Theorem 2.1

\newlabelthm:gamma-con0 Suppose that $\Omega$ is a Lipschitz bounded domain in $\mathbb{R}^d$. $1<p<\infty$ is a constant. $K,R$ are two kernel functions satisfying (K1)-(K3), with $\sigma_R$ given by eq:sigmaR. $\{\delta_n\}$ is a sequence of positive constants tending to 0 as $n\rightarrow\infty where $F_n,F$ are defined as eq:dis-fun,eq:con-fun.

Theorems & Definitions (35)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Definition 4.1: $\varGamma$-convergence
  • Lemma 4.2: Convergence of minimizers
  • ...and 25 more