Homogenization for nonlocal problems with smooth kernels

Monia Capanna; Jean C. Nakasato; Marcone C. Pereira; Julio D. Rossi

doi:10.3934/dcds.2020385

Article Contents

2021, Volume 41, Issue 6: 2777-2808. Doi: 10.3934/dcds.2020385

This issue Previous Article Forward untangling and applications to the uniqueness problem for the continuity equation Next Article Properties of multicorrelation sequences and large returns under some ergodicity assumptions

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
^* Corresponding author: Julio D. Rossi

Received: April 30, 2020

Revised: September 30, 2020

Early access: November 2020

Published: June 2021

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)

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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.

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Mathematics Subject Classification: Primary: 45A05, 45C05; Secondary: 45M05.

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