In this work we study the asymptotic behavior of a class of quasilinear elliptic problems posed in a domain perforated by epsilon-periodic holes of size epsilon. The quasilinear equations present a nonlinear singular lower order term f?(u(epsilon)), where u(epsilon) is the solution of the problem at epsilon-level, ? is a continuous function singular in zero and f a function whose summability depends on the growth of ? near its singularity. We prescribe a nonlinear Robin condition on the boundary of the holes contained in Omega and a homogeneous Dirichlet condition on the exterior boundary. The particular case of a Neumann boundary condition on the holes is already new.The main tool in the homogenization process consists in proving a suitable convergence result, which shows that the gradient of u(epsilon) behaves like that of the solution of a suitable linear problem associated with a weak cluster point of the sequence {u(epsilon)}, as epsilon -> 0. This allows us not only to pass to the limit in the quasilinear term, but also to study the singular term near its singularity, via an accurate a priori estimate. We also get a corrector result for our problem.The main novelty of this work is that for the first time the unfolding method is used to treat a singular term as f?(u(epsilon)). This plays an essential role in order to get an almost everywhere convergence of the solution u(epsilon), needed in the study the asymptotic behavior of the problem.

Homogenization of a class of singular elliptic problems in perforated domains

Sara Monsurrò;Federica Raimondi
2018-01-01

Abstract

In this work we study the asymptotic behavior of a class of quasilinear elliptic problems posed in a domain perforated by epsilon-periodic holes of size epsilon. The quasilinear equations present a nonlinear singular lower order term f?(u(epsilon)), where u(epsilon) is the solution of the problem at epsilon-level, ? is a continuous function singular in zero and f a function whose summability depends on the growth of ? near its singularity. We prescribe a nonlinear Robin condition on the boundary of the holes contained in Omega and a homogeneous Dirichlet condition on the exterior boundary. The particular case of a Neumann boundary condition on the holes is already new.The main tool in the homogenization process consists in proving a suitable convergence result, which shows that the gradient of u(epsilon) behaves like that of the solution of a suitable linear problem associated with a weak cluster point of the sequence {u(epsilon)}, as epsilon -> 0. This allows us not only to pass to the limit in the quasilinear term, but also to study the singular term near its singularity, via an accurate a priori estimate. We also get a corrector result for our problem.The main novelty of this work is that for the first time the unfolding method is used to treat a singular term as f?(u(epsilon)). This plays an essential role in order to get an almost everywhere convergence of the solution u(epsilon), needed in the study the asymptotic behavior of the problem.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11386/4811557
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