American Institute of Mathematical Sciences

June  2021, 41(6): 2777-2808. doi: 10.3934/dcds.2020385

Homogenization for nonlocal problems with smooth kernels

 1 CONICET and Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Ciudad Universitaria, Pabellon I, (1428), Buenos Aires, Argentina 2 Dpto. de Matemática, ICMC, Universidade de São Paulo, Avenida Trabalhador São-Carlense, 400, São Carlos - SP, Brazil 3 Dpto. de Matemática Aplicada, IME, Universidade de São Paulo, Rua do Matão 1010, São Paulo - SP, Brazil

* Corresponding author: Julio D. Rossi

Received  May 2020 Revised  October 2020 Published  June 2021 Early access  November 2020

Fund Project: The first and last authors (MC and JDR) are partially supported by CONICET grant PIP GI No 11220150100036CO (Argentina), UBACyT grant 20020160100155BA (Argentina), Project MTM2015-70227-P (Spain).
The third author (MCP) has been partially supported by CNPq 303253/2017-7 and FAPESP 2020/04813-0 (Brazil).
The second author (JCN) supported by CAPES - INCTmat grant 465591/2014-0 (Brazil)

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.

Citation: Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2777-2808. doi: 10.3934/dcds.2020385
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