On the behaviour of solutions to the Dirichlet problem for the p(x)-Laplacian when p(x) goes to 1 in a subdomain
In this paper we prove a stability result for some classes of elliptic problems involving variable exponents. More precisely, we consider the Dirichlet problem for an elliptic equation in a domain, which is the p-Laplacian equation, -div(∇up-2∇u) = f, in a subdomain Ω1 of Ω and the Laplace equation,...
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| Publicado: |
2012
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_08934983_v25_n1-2_p53_Mercaldo http://hdl.handle.net/20.500.12110/paper_08934983_v25_n1-2_p53_Mercaldo |
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| Sumario: | In this paper we prove a stability result for some classes of elliptic problems involving variable exponents. More precisely, we consider the Dirichlet problem for an elliptic equation in a domain, which is the p-Laplacian equation, -div(∇up-2∇u) = f, in a subdomain Ω1 of Ω and the Laplace equation, -Δu = f, in its complementary (that is, our equation involves the so-called p(x)-Laplacian with a discontinuous exponent). We assume that the right-hand side f belongs to L∞(Ω). For this problem, we study the behaviour of the solutions as p goes to 1, showing that they converge to a function u, which is almost everywhere finite when the size of the datum f is small enough. Moreover, we prove that this u is a solution of a limit problem involving the 1-Laplacian operator in Ω1. We also discuss uniqueness under a favorable geometry. |
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