An inhomogeneous singular perturbation problem for the p(x)-Laplacian Dedicated to our dear friend and colleague Juan Luis Vázquez on the occasion of his 70th birthday

In this paper we study the following singular perturbation problem for the pϵ(x)-Laplacian: Δpϵ (x)uϵ:=div(|∇uϵ(x)|pϵ (x)-2∇ uϵ)=βϵ(uϵ)+fϵ,uϵ≥0, (Pϵ(fϵ, pϵ)) where ϵ>0, βϵ(s)=1/ϵβ(s/ϵ), with β a Lipschitz function satisfying β>0 in (0,1), β≡0 outside (0,1) and ∫β(s)ds=M. The functions...

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Autores principales: Lederman, C., Wolanski, N.
Formato: JOUR
Materias:
Free boundary problem
Singular perturbation
Variable exponent spaces
Boundary value problems
Free-boundary problems
Lipschitz functions
Lipschitz regularity
P (x)-Laplacian
Singular perturbation problems
Singular perturbations
Uniformly bounded
Variable exponents
Laplace transforms
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_0362546X_v138_n_p300_Lederman
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id todo:paper_0362546X_v138_n_p300_Lederman
record_format dspace
spelling todo:paper_0362546X_v138_n_p300_Lederman2023-10-03T15:27:11Z An inhomogeneous singular perturbation problem for the p(x)-Laplacian Dedicated to our dear friend and colleague Juan Luis Vázquez on the occasion of his 70th birthday Lederman, C. Wolanski, N. Free boundary problem Singular perturbation Variable exponent spaces Boundary value problems Free-boundary problems Lipschitz functions Lipschitz regularity P (x)-Laplacian Singular perturbation problems Singular perturbations Uniformly bounded Variable exponents Laplace transforms In this paper we study the following singular perturbation problem for the pϵ(x)-Laplacian: Δpϵ (x)uϵ:=div(|∇uϵ(x)|pϵ (x)-2∇ uϵ)=βϵ(uϵ)+fϵ,uϵ≥0, (Pϵ(fϵ, pϵ)) where ϵ>0, βϵ(s)=1/ϵβ(s/ϵ), with β a Lipschitz function satisfying β>0 in (0,1), β≡0 outside (0,1) and ∫β(s)ds=M. The functions uϵ, fϵ and pϵ are uniformly bounded. We prove uniform Lipschitz regularity, we pass to the limit (ϵ→0) and we show that, under suitable assumptions, limit functions are weak solutions to the free boundary problem: u≥0 and {Δp(x)u = f in {u>0}u=0,|∇u|=λ ∗(x)on ∂{u>0} (P(f, p, λ∗)) with λ∗ (x)=(p(x)/p(x)-1 M)1/p(x), p = lim pϵ and f = lim fϵ. In Lederman and Wolanski (submitted) we prove that the free boundary of a weak solution is a C1,α surface near flat free boundary points. This result applies, in particular, to the limit functions studied in this paper. © 2015 Elsevier Ltd. All rights reserved. Fil:Lederman, C. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Wolanski, N. 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_0362546X_v138_n_p300_Lederman
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Free boundary problem
Singular perturbation
Variable exponent spaces
Boundary value problems
Free-boundary problems
Lipschitz functions
Lipschitz regularity
P (x)-Laplacian
Singular perturbation problems
Singular perturbations
Uniformly bounded
Variable exponents
Laplace transforms
spellingShingle Free boundary problem
Singular perturbation
Variable exponent spaces
Boundary value problems
Free-boundary problems
Lipschitz functions
Lipschitz regularity
P (x)-Laplacian
Singular perturbation problems
Singular perturbations
Uniformly bounded
Variable exponents
Laplace transforms
Lederman, C.
Wolanski, N.
An inhomogeneous singular perturbation problem for the p(x)-Laplacian Dedicated to our dear friend and colleague Juan Luis Vázquez on the occasion of his 70th birthday
topic_facet Free boundary problem
Singular perturbation
Variable exponent spaces
Boundary value problems
Free-boundary problems
Lipschitz functions
Lipschitz regularity
P (x)-Laplacian
Singular perturbation problems
Singular perturbations
Uniformly bounded
Variable exponents
Laplace transforms
description In this paper we study the following singular perturbation problem for the pϵ(x)-Laplacian: Δpϵ (x)uϵ:=div(|∇uϵ(x)|pϵ (x)-2∇ uϵ)=βϵ(uϵ)+fϵ,uϵ≥0, (Pϵ(fϵ, pϵ)) where ϵ>0, βϵ(s)=1/ϵβ(s/ϵ), with β a Lipschitz function satisfying β>0 in (0,1), β≡0 outside (0,1) and ∫β(s)ds=M. The functions uϵ, fϵ and pϵ are uniformly bounded. We prove uniform Lipschitz regularity, we pass to the limit (ϵ→0) and we show that, under suitable assumptions, limit functions are weak solutions to the free boundary problem: u≥0 and {Δp(x)u = f in {u>0}u=0,|∇u|=λ ∗(x)on ∂{u>0} (P(f, p, λ∗)) with λ∗ (x)=(p(x)/p(x)-1 M)1/p(x), p = lim pϵ and f = lim fϵ. In Lederman and Wolanski (submitted) we prove that the free boundary of a weak solution is a C1,α surface near flat free boundary points. This result applies, in particular, to the limit functions studied in this paper. © 2015 Elsevier Ltd. All rights reserved.
format JOUR
author Lederman, C.
Wolanski, N.
author_facet Lederman, C.
Wolanski, N.
author_sort Lederman, C.
title An inhomogeneous singular perturbation problem for the p(x)-Laplacian Dedicated to our dear friend and colleague Juan Luis Vázquez on the occasion of his 70th birthday
title_short An inhomogeneous singular perturbation problem for the p(x)-Laplacian Dedicated to our dear friend and colleague Juan Luis Vázquez on the occasion of his 70th birthday
title_full An inhomogeneous singular perturbation problem for the p(x)-Laplacian Dedicated to our dear friend and colleague Juan Luis Vázquez on the occasion of his 70th birthday
title_fullStr An inhomogeneous singular perturbation problem for the p(x)-Laplacian Dedicated to our dear friend and colleague Juan Luis Vázquez on the occasion of his 70th birthday
title_full_unstemmed An inhomogeneous singular perturbation problem for the p(x)-Laplacian Dedicated to our dear friend and colleague Juan Luis Vázquez on the occasion of his 70th birthday
title_sort inhomogeneous singular perturbation problem for the p(x)-laplacian dedicated to our dear friend and colleague juan luis vázquez on the occasion of his 70th birthday
url http://hdl.handle.net/20.500.12110/paper_0362546X_v138_n_p300_Lederman
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