In this paper, we analyse nonlocal equations in perforated domains. We consider nonlocal problems of the form with x in a perforated domain. Here J is a nonsingular kernel. We think about 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 latter case we impose that u vanishes in the holes but integrate in the whole ℝN (B = ℝN) and in the former we just consider integrals in ℝN minus the holes (B = ℝN \\ ω\\ωϵ). Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of has a weak limit, weakly∗ in L∞(ω), we analyse the limit as ϵ → 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 the order of the size of the typical cell (that gives the period). In addition, in this periodic case, we also study the behaviour of these nonlocal problems when we rescale the kernel in order to approximate local PDE problems. © Royal Society of Edinburgh 2019.
|Título:||Nonlocal problems in perforated domains|
|Autor:||Pereira, M.C.; Rossi, J.D.|
|Filiación:||Dpto. de Matemática Aplicada, IME, Universidade de São Paulo, Rua do Matão 1010, São Paulo - SP, Brazil|
Dpto. de Matemáticas, FCEyN, Universidad de Buenos Aires, Ciudad Universitaria Pab 1, Buenos Aires, 1428, Argentina
|Palabras clave:||Dirichlet problem; Neumann problem; nonlocal equations; perforated domains|
|Título revista:||Proceedings of the Royal Society of Edinburgh Section A: Mathematics|
|Título revista abreviado:||Proc. R. Soc. Edinburgh Sect. A Math.|