Homogenization for nonlocal problems with smooth kernels

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ A_n \cup B_n $\end{document}</tex-math></inline-formula> and we have three different smooth kernels, one that controls the jumps from <inline-formula><tex-math id="M2">\begin{document}$ A_n $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M3">\begin{document}$ A_n $\end{document}</tex-math></inline-formula>, a second one that controls the jumps from <inline-formula><tex-math id="M4">\begin{document}$ B_n $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M5">\begin{document}$ B_n $\end{document}</tex-math></inline-formula> and the third one that governs the interactions between <inline-formula><tex-math id="M6">\begin{document}$ A_n $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ B_n $\end{document}</tex-math></inline-formula>. Assuming that <inline-formula><tex-math id="M8">\begin{document}$ \chi_{A_n} (x) \to X(x) $\end{document}</tex-math></inline-formula> weakly-* in <inline-formula><tex-math id="M9">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula> (and then <inline-formula><tex-math id="M10">\begin{document}$ \chi_{B_n} (x) \to (1-X)(x) $\end{document}</tex-math></inline-formula> weakly-* in <inline-formula><tex-math id="M11">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula>) as <inline-formula><tex-math id="M12">\begin{document}$ n \to \infty $\end{document}</tex-math></inline-formula> we show that there is an homogenized limit system in which the three kernels and the limit function <inline-formula><tex-math id="M13">\begin{document}$ X $\end{document}</tex-math></inline-formula> appear. We deal with both Neumann and Dirichlet boundary conditions. Moreover, we also provide a probabilistic interpretation of our results.