Homogenization for nonlocal problems with smooth kernels

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 An ∪ Bn and we have three different smooth kernels, one that controls the jumps from An to An, a second one that controls the jumps from Bn to Bn and the third one that governs the interactions between An and Bn. Assuming that χA n (x) → X(x) weakly-* in L ∞ (and then χB n (x) → (1 − X)(x) weakly-* in L ∞ ) as n → ∞ 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.

Homogenization for nonlocal evolution problems with three different smooth kernels

“…Observe that w n is a solution to (2.43) By Lemma 4.1 in [9] we know that there exists a positive constant c (independent of n) such that…”

“…By Lemma 4.1 in [9] it holds that there exists a constant c 1 (independent of n) such that E n (w n ) ≥ 2c 1 w n (t, •) 4 The Dirichlet case.…”

“…Our main goal in this paper is to study the homogenization that occurs when one deals with nonlocal evolution problems with different non-singular kernels that act in different domains. This paper is a natural continuation of [9] where the stationary case was studied.…”