An inhomogeneous singular perturbation problem for the p(x)-Laplacian. (June 2016)

• Record Type: Journal Article
• Authors: Lederman, Claudia; Wolanski, Noemi
• Is Part Of: Nonlinear analysis. Volume 138(2016)
• Page Start: 300
• Page End: 325
• Publication Date: 2016-06
• Subjects: 35R35 -- 35B65 -- 35J60 -- 35J70; Free boundary problem -- Variable exponent spaces -- Singular perturbation; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.09.026
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we study the following singular perturbation problem for the p ε ( x ) -Laplacian: ( P ε ( f ε, p ε ) ) Δ p ε ( x ) u ε : = div ( | ∇ u ε ( x ) | p ε ( x ) − 2 ∇ u ε ) = β ε ( u ε ) + f ε, u ε ≥ 0, where ε > 0, β ε ( s ) = 1 ε β ( s ε ), with β a Lipschitz function satisfying β > 0 in ( 0, 1 ), β ≡ 0 outside ( 0, 1 ) and ∫ β ( s ) d s = M . The functions u ε, f ε and p ε 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 ( P ( f, p, λ ∗ ) ) { Δ p ( x ) u = f in { u > 0 } u = 0, | ∇ u | = λ ∗ ( x ) on ∂ { u > 0 } with λ ∗ ( x ) = ( p ( x ) p ( x ) − 1 M ) 1 / p ( x ), p = lim p ε and f = lim f ε . In Lederman and Wolanski (submitted) we prove that the free boundary of a weak solution is a C 1, α surface near flat free boundary points. This result applies, in particular, to the limit functions studied in this paper.