Potential Analysis

, Volume 48, Issue 3, pp 361–373 | Cite as

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 Ju (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, uψ 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.

Keywords

Perforated domains Nonlocal equations Neumann problem Dirichlet problem 

Mathematics Subject Classification (2010)

45A05 45M05 49J40 

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Notes

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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© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Dpto. de Matemática Aplicada, IMEUniversidade de São PauloSão PauloBrazil
  2. 2.Dpto. de Matemáticas, FCEyNUniversidad de Buenos AiresBuenos AiresArgentina

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