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Homogenization for nonlocal problems with smooth kernels

  • * Corresponding author: Julio D. Rossi

    * Corresponding author: Julio D. Rossi

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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  • 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.

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


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