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An Obstacle Problem for Nonlocal Equations in Perforated Domains

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Abstract

In this paper we analyze the behavior of solutions to a nonlocal equation of the form J ∗ u (x) − u (x) = f (x) in a perforated domain Ω ∖ A 𝜖 with u = 0 in \(A^{\epsilon } \cup {\Omega }^{c}\) and an obstacle constraint, u ≥ ψ in Ω ∖ A 𝜖. We show that, assuming that the characteristic function of the domain Ω ∖ A 𝜖 verifies \(\chi _{\epsilon } \rightharpoonup \mathcal {X}\) weakly ∗ in \(L^{\infty }({\Omega })\), there exists a weak limit of the solutions u 𝜖 and we find the limit problem that is satisfied in the limit. When \(\mathcal {X} \not \equiv 1\) in this limit problem an extra term appears in the equation as well as a modification of the obstacle constraint inside the domain.

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Acknowledgements

The first author (MCP) is partially supported by CNPq 302960/2014-7 and 471210/2013-7, FAPESP 2015/17702-3 (Brazil) and the second author (JDR) by MINCYT grant MTM2016-68210 (Spain).

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Authors and Affiliations

  1. Dpto. de Matemática Aplicada, IME, Universidade de São Paulo, Rua do Matão 1010, São Paulo, SP, Brazil

    Marcone C. Pereira

  2. Dpto. de Matemáticas, FCEyN, Universidad de Buenos Aires, Ciudad Universitaria Pab 1, 1428, Buenos Aires, Argentina

    Julio D. Rossi

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  1. Marcone C. Pereira
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  2. Julio D. Rossi
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Correspondence to Marcone C. Pereira.

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Pereira, M.C., Rossi, J.D. An Obstacle Problem for Nonlocal Equations in Perforated Domains. Potential Anal 48, 361–373 (2018). https://doi.org/10.1007/s11118-017-9639-5

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