Homogenization for nonlocal problems with smooth kernels

TL;DR

This paper studies the homogenization of nonlocal equations with multiple smooth kernels in divided domains, deriving limit systems that incorporate the kernels and a limiting characteristic function, with applications to boundary conditions and probabilistic interpretations.

Contribution

It introduces a homogenization framework for nonlocal problems with multiple kernels and domain partitions, including boundary conditions and probabilistic insights.

Findings

Homogenized limit system involving three kernels and a limit function X

Results apply to both Neumann and Dirichlet boundary conditions

Provides a probabilistic interpretation of the homogenization process

Abstract

In this paper we consider the homogenization problem for a nonlocal equation that involve different smooth kernels. We assume that the spacial domain is divided into a sequence of two subdomains $A_n \cup B_n$ and we have three different smooth kernels, one that controls the jumps from $A_n$ to $A_n$, a second one that controls the jumps from $B_n$ to $B_n$ and the third one that governs the interactions between $A_n$ and $B_n$. Assuming that $\chi_{A_n} (x) \to X(x)$ weakly-* in $L^\infty$ (and then $\chi_{B_n} (x) \to (1-X)(x)$ weakly-* in $L^\infty$) as $n \to \infty$ we show that there is an homogenized limit system in which the three kernels and the limit function $X$ appear. We deal with both Neumann and Dirichlet boundary conditions. Moreover, we also provide a probabilistic interpretation of our results.