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Periodic Homogenization of Local/Nonlocal Systems

Marcone C. Pereira, Luiza C. Rosa da Silva, Julio D. Rossi

Abstract

In this paper, we study the homogenization of elliptic equations that combine a local part, given by the Laplacian with Neumann boundary conditions, and its nonlocal version, defined through an integral operator with a smooth kernel. These two components are coupled through an additional nonlocal operator also given by a smooth kernel. We consider a sequence of partitions of a fixed spatial domain into two regions - local and nonlocal - which are periodically distributed in space (with one of the regions consisting of small, periodically arranged holes). Depending on the relative location of the local and nonlocal regions, we obtain qualitatively different limit behaviors. When the local part of the equation is confined to the small periodic holes, the sequence of solutions converges to the unique solution of a limit system in which the local component vanishes, while the nonlocal part persists and splits into two distinct components. On the other hand, when the local part of the problem lies outside the holes, the limit system exhibits a homogenized local diffusion operator coupled with a nonlocal equation. Finally, we analyze an intermediate regime in which only part of the local diffusion survives in the limit, by considering configurations consisting of parallel thin strips instead of holes.

Periodic Homogenization of Local/Nonlocal Systems

Abstract

In this paper, we study the homogenization of elliptic equations that combine a local part, given by the Laplacian with Neumann boundary conditions, and its nonlocal version, defined through an integral operator with a smooth kernel. These two components are coupled through an additional nonlocal operator also given by a smooth kernel. We consider a sequence of partitions of a fixed spatial domain into two regions - local and nonlocal - which are periodically distributed in space (with one of the regions consisting of small, periodically arranged holes). Depending on the relative location of the local and nonlocal regions, we obtain qualitatively different limit behaviors. When the local part of the equation is confined to the small periodic holes, the sequence of solutions converges to the unique solution of a limit system in which the local component vanishes, while the nonlocal part persists and splits into two distinct components. On the other hand, when the local part of the problem lies outside the holes, the limit system exhibits a homogenized local diffusion operator coupled with a nonlocal equation. Finally, we analyze an intermediate regime in which only part of the local diffusion survives in the limit, by considering configurations consisting of parallel thin strips instead of holes.
Paper Structure (10 sections, 11 theorems, 317 equations)

This paper contains 10 sections, 11 theorems, 317 equations.

Table of Contents

  1. Introduction
  2. Description of mixed local and nonlocal equations and statements of the main results
  3. Local in Holes
  4. Nonlocal in Holes
  5. Local in Strips
  6. Existence and uniqueness of solutions to the system \ref{['local']}--\ref{['nonlocal']}
  7. Existence and uniqueness
  8. Local in the Holes
  9. NonLocal in Holes
  10. Strip Periodic Case

Key Result

Theorem 2.1

Let $(u_n,v_n)\in W_n$ be the family of solutions to hlocal--hnl, where $A_n$ is the union of holes holes. Then, as $n\to\infty$, the following convergences hold: and the limit pair $(u,v)\in L^2(\Omega)\times L^2(\Omega)$ satisfies the compatibility condition Moreover, the limit $(u,v)\in L^2(\Omega)\times L^2(\Omega)$ is uniquely characterized as the weak solution of the following homogenized

Theorems & Definitions (25)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3: Nonlocal in Holes
  • Theorem 2.4: Corrector for Nonlocal in Holes
  • Theorem 2.5
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof
  • Lemma 4.1: Uniform nonlocal Poincaré inequality on the perforated domains
  • ...and 15 more