Nonlocal problems in perforated domains

Abstract

In this paper we analyze nonlocal equations in perforated domains. We consider nonlocal problems of the form f (x) = B J(x − y)(u(y) − u(x))dy with x in a perforated domain Ω ⊂ Ω. Here J is a non-singular kernel. We think about Ω as a fixed set Ω from where we have removed a subset that we call the holes. We deal both with the Neumann and Dirichlet conditions in the holes and assume a Dirichlet condition outside Ω. In the later case we impose that u vanishes in the holes but integrate in the whole R N (B = R N ) and in the former we just consider integrals in R N minus the holes (B = R N \ (Ω \ Ω )). Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of Ω has a weak limit, χ X weakly * in L ∞ (Ω), we analyze the limit as → 0 of the solutions to the nonlocal problems proving that there is a nonlocal limit problem. In the case in which the holes are periodically removed balls we obtain that the critical radius is of order of the size of the typical cell (that gives the period). In addition, in this periodic case, we also study the behavior of these nonlocal problems when we rescale the kernel in order to approximate local PDE problems.

Nonlocal evolution equations in perforated domains

“…Let us emphasize that these present results are in agreement with the previous ones from the work of Pereira and Rossi. 1 Indeed, this present work is a natural continuation of the aforementioned work 1 where stationary nonlocal equations in perforated domains have been studied. It is known that strong convergence in this context is not possible in general.…”

“…See, for instance, the works of Pereira and Rossi. 1,32 In some sense, nonlocal approximations to local problems under singular perturbations have to be done in a very careful way. This paper is organized as follows.…”

Homogenization for nonlocal problems with smooth kernels

“…By (23) we can conclude that the limit of the right hand side of (33) converges to 0 as K → ∞. Therefore (29) is proved.…”

“…We refer to [3,9,25] as general references for the subject. For other kinds of homogenization for pure nonlocal problems with one kernel we refer to [22,23,21]. For homogenization results for equations with a singular kernel we refer to [7,24,27] and references therein.…”

“…Both cases, X(x) ≡ 0 or X(x) ≡ 1, can be seen as a kind of performing weakly perforations in Ω setting a prevailing situation at the limit equation, see [23].…”