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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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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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 |
_version_ |
1768546215476068352 |