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

• Published in: Nonlinear analysis 2016-06, Vol.138, p.300-325
• Main Authors: Lederman, Claudia, Wolanski, Noemi
• Format: Article
• Language: English
• Subjects: Boundaries; Flats; Free boundaries; Mathematical analysis; Nonlinearity; Regularity; Singular perturbation
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2015.09.026

In this paper we study the following singular perturbation problem for the p sub( epsilon )(x)p epsilon (x)-Laplacian: equation(P sub( epsilon )(f super( epsilon ),p sub( epsilon ))P epsilon (f epsilon ,p epsilon ) Turn MathJax on ) View the MathML source Delta p epsilon (x)u epsilon :=div(|(grad)u epsilon (x) |p epsilon (x)-2(grad)u epsilon )= beta epsilon (u epsilon )+f epsilon ,u epsilon greater than or equal to 0, Turn MathJax on where epsilon >0 epsilon >0, View the MathML source beta epsilon (s)=1 epsilon beta (s epsilon ), with beta beta a Lipschitz function satisfying beta >0 beta >0 in (0,1)(0,1), beta identical with 0 beta identical with 0 outside (0,1)(0,1) and View the MathML source [int] beta (s)ds=M. The functions u super( epsilon )u epsilon , f super( epsilon )f epsilon and p sub( epsilon )p epsilon are uniformly bounded. We prove uniform Lipschitz regularity, we pass to the limit ( epsilon arrow right 0)( epsilon arrow right 0) and we show that, under suitable assumptions, limit functions are weak solutions to the free boundary problem: u greater than or equal to 0u greater than or equal to 0 and equation(P(f,p, lambda super([lowast]))P(f,p, lambda [lowast]) Turn MathJax on ) View the MathML source { Delta p(x)u=fin {u>0}u=0,|(grad)u|= lambda [lowast](x)on partial differential {u>0} Turn MathJax on with View the MathML source lambda [lowast](x)=(p(x)p(x)-1M)1/p(x), p=limp sub( epsilon )p=limp epsilon and f=limf super( epsilon )f=limf epsilon . In Lederman and Wolanski (submitted) we prove that the free boundary of a weak solution is a C super(1, alpha )C1, alpha surface near flat free boundary points. This result applies, in particular, to the limit functions studied in this paper.