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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Autor principal: Rossi, Julio Daniel
Publicado: 2012
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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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling paper:paper_08934983_v25_n1-2_p53_Mercaldo2023-06-08T15:47:32Z On the behaviour of solutions to the Dirichlet problem for the p(x)-Laplacian when p(x) goes to 1 in a subdomain Rossi, Julio Daniel 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. Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2012 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
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
description 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.
author Rossi, Julio Daniel
spellingShingle Rossi, Julio Daniel
On the behaviour of solutions to the Dirichlet problem for the p(x)-Laplacian when p(x) goes to 1 in a subdomain
author_facet Rossi, Julio Daniel
author_sort Rossi, Julio Daniel
title On the behaviour of solutions to the Dirichlet problem for the p(x)-Laplacian when p(x) goes to 1 in a subdomain
title_short On the behaviour of solutions to the Dirichlet problem for the p(x)-Laplacian when p(x) goes to 1 in a subdomain
title_full On the behaviour of solutions to the Dirichlet problem for the p(x)-Laplacian when p(x) goes to 1 in a subdomain
title_fullStr On the behaviour of solutions to the Dirichlet problem for the p(x)-Laplacian when p(x) goes to 1 in a subdomain
title_full_unstemmed On the behaviour of solutions to the Dirichlet problem for the p(x)-Laplacian when p(x) goes to 1 in a subdomain
title_sort on the behaviour of solutions to the dirichlet problem for the p(x)-laplacian when p(x) goes to 1 in a subdomain
publishDate 2012
url 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
work_keys_str_mv AT rossijuliodaniel onthebehaviourofsolutionstothedirichletproblemforthepxlaplacianwhenpxgoesto1inasubdomain
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