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

2015/10/01 by Claudia Lederman, Lederman, Claudia, Noemí Wolanski +2
Computer Science · Mathematics · #35B65 #35J60 #35J70 #35R35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics #math.AP #msc:35B65 #msc:35J60 #msc:35J70 #msc:35R35

paper · pdf · doi:10.48550/arxiv.1510.00316

Nonlinear Analysis TM&A, to appear

arxiv created 2015/10/01 · openalex publication_date 2015/10/01 · arxiv updated 2015/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

In this paper we study the following singular perturbation problem for the p_ε(x)-Laplacian: Δ_pε(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 \over ε β(s \over ε), with β a Lipschitz function satisfying β>0 in (0,1), β≡ 0 outside (0,1) and ∫ β(s) ds=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 \begincases Δ_p(x)u= f in \u>0\
u=0, |∇ u| = λ^*(x) on ∂\u>0\ \endcases with λ^*(x)=((p(x))/(p(x)-1) M)^1/p(x), p=lim p_ε and f=lim f_ε. In \citeLW4 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.

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