Nonlocal problems in perforated domains

Autor(es)

Pereira, Marcone C. ^[1] ; Rossi, Julio D. ^[2]

Afiliação do(s) autor(es)

^[1] Univ Sao Paulo, IME, Dept Matemat Aplicada, Rua Matao 1010, Sao Paulo, SP - Brazil

^[2] Univ Buenos Aires, FCEyN, Dept Matemat, Ciudad Univ Pab 1, RA-1428 Buenos Aires, DF - Argentina

• Tipo de documento: Artigo Científico
• Fonte: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS; v. 150, n. 1, p. 305-340, FEB 2020.
• Citações Web of Science: 0

Resumo

In this paper, we analyse 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 O. O. Here J is a nonsingular kernel. We think about O as a fixed set O 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 O. 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 \textbackslash{} (O \textbackslash{} O)). Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of O has a weak limit,. X weakly{*} in L8( O), we analyse 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 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. (AU)

• Processo FAPESP: 15/17702-3 - Problemas de obstáculo para evolução não-local em espaço e tempo
• Beneficiário: Marcone Corrêa Pereira
• Modalidade de apoio: Bolsas no Exterior - Pesquisa
• Processo FAPESP: 17/02630-2 - Análise assintótica de equações diferenciais e integrais
• Beneficiário: Marcone Corrêa Pereira
• Modalidade de apoio: Auxílio à Pesquisa - Regular