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Abstract:

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.

Registro:

Documento: Artículo
Título: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
Autor:Lederman, C.; Wolanski, N.
Filiación:IMAS-CONICET, Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, (1428), Argentina
Palabras clave: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
Año:2016
Volumen:138
Página de inicio:300
Página de fin:325
DOI: http://dx.doi.org/10.1016/j.na.2015.09.026
Título revista:Nonlinear Analysis, Theory, Methods and Applications
Título revista abreviado:Nonlinear Anal Theory Methods Appl
ISSN:0362546X
CODEN:NOAND
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0362546X_v138_n_p300_Lederman

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Citas:

---------- APA ----------
Lederman, C. & Wolanski, N. (2016) . 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. Nonlinear Analysis, Theory, Methods and Applications, 138, 300-325.
http://dx.doi.org/10.1016/j.na.2015.09.026
---------- CHICAGO ----------
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" . Nonlinear Analysis, Theory, Methods and Applications 138 (2016) : 300-325.
http://dx.doi.org/10.1016/j.na.2015.09.026
---------- MLA ----------
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" . Nonlinear Analysis, Theory, Methods and Applications, vol. 138, 2016, pp. 300-325.
http://dx.doi.org/10.1016/j.na.2015.09.026
---------- VANCOUVER ----------
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. Nonlinear Anal Theory Methods Appl. 2016;138:300-325.
http://dx.doi.org/10.1016/j.na.2015.09.026