Luiza Camile Rosa da Silva

Universidade de São Paulo

Title: Homogenization of a local/nonlocal problem with periodic holes

Abstract: Our main goal is to study the homogenization that occurs when one deals

with the mix problem that involves the Laplacian and Nonlocal Integral Form operators with two different non-singular kernels that act in different domains $A_n,B_n$, which is described by a system where the nonlocal terms which involved kernels are radial probability densities and the local laplacian term. Our ambient space $\Omega$ is divided into two disjoint $A_n$ and $B_n$ sets. This study introduce two different settings for the domain $\Omega$, that is divided as $A_n$ being the union of finite, periodic and disjoint balls with the same radium $r_n=1/n$, and the opposite case, where $B_n$ is the union of finite balls described above.

We have a weak convergence of the characteristic functions of $A_n$ as $n\to \infty$ to a bounded function $X:\Omega \to [0,1]$ and we will prove that, passing to the limit in \eqref{local}-\eqref{nonlocal}, the first case tells us that the homogenized equation (in terms of $X$) gets the disappearance of local term, while in the second case the local term will survive. Joint work with Marcone Pereira and Julio Rossi.