An Obstacle Problem for Nonlocal Equations in Perforated Domains |
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Authors: | Marcone C Pereira Julio D Rossi |
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Institution: | 1.Dpto. de Matemática Aplicada, IME,Universidade de S?o Paulo,S?o Paulo,Brazil;2.Dpto. de Matemáticas, FCEyN,Universidad de Buenos Aires,Buenos Aires,Argentina |
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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 ?? with u = 0 in \(A^{\epsilon } \cup {\Omega }^{c}\) and an obstacle constraint, u ≥ ψ in Ω ? A ?? . We show that, assuming that the characteristic function of the domain Ω ? A ?? verifies \(\chi _{\epsilon } \rightharpoonup \mathcal {X}\) weakly ? in \(L^{\infty }({\Omega })\), there exists a weak limit of the solutions u ?? 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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