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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Detalles Bibliográficos
Autores principales: Mercaldo, A., Rossi, J.D., De León, S.S., Trombetti, C.
Formato: JOUR
Acceso en línea: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
id todo:paper_08934983_v25_n1-2_p53_Mercaldo
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spelling todo:paper_08934983_v25_n1-2_p53_Mercaldo2023-10-03T15:41:45Z On the behaviour of solutions to the Dirichlet problem for the p(x)-Laplacian when p(x) goes to 1 in a subdomain Mercaldo, A. Rossi, J.D. De León, S.S. Trombetti, C. 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar 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.
format JOUR
author Mercaldo, A.
Rossi, J.D.
De León, S.S.
Trombetti, C.
spellingShingle Mercaldo, A.
Rossi, J.D.
De León, S.S.
Trombetti, C.
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 Mercaldo, A.
Rossi, J.D.
De León, S.S.
Trombetti, C.
author_sort Mercaldo, A.
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
url http://hdl.handle.net/20.500.12110/paper_08934983_v25_n1-2_p53_Mercaldo
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