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- W4214967508 abstract "In this paper we study the following singular perturbation problem for the $p_varepsilon(x)$-Laplacian: [ Delta_{p_varepsilon(x)}u^varepsilon:=mbox{div}(|nabla u^varepsilon(x)|^{p_varepsilon(x)-2}nabla u^varepsilon)={beta}_{varepsilon}(u^varepsilon)+f_varepsilon, quad u^varepsilongeq 0, ] where $varepsilon>0$, ${beta}_{varepsilon}(s)={1 over varepsilon} beta({s over varepsilon})$, with $beta$ a Lipschitz function satisfying $beta>0$ in $(0,1)$, $betaequiv 0$ outside $(0,1)$ and $int beta(s), ds=M$. The functions $u^varepsilon$, $f_varepsilon$ and $p_varepsilon$ are uniformly bounded. We prove uniform Lipschitz regularity, we pass to the limit $(varepsilonto 0)$ and we show that, under suitable assumptions, limit functions are weak solutions to the free boundary problem: $uge0$ and [ begin{cases} Delta_{p(x)}u= f & mbox{in }{u>0} u=0, |nabla u| = lambda^*(x) & mbox{on }partial{u>0} end{cases} ] with $lambda^*(x)=Big(frac{p(x)}{p(x)-1},MBig)^{1/p(x)}$, $p=lim p_varepsilon$ and $f=lim f_varepsilon$. In cite{LW4} we prove that the free boundary of a weak solution is a $C^{1,alpha}$ surface near flat free boundary points. This result applies, in particular, to the limit functions studied in this paper." @default.
- W4214967508 created "2022-03-05" @default.
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- W4214967508 date "2015-10-01" @default.
- W4214967508 modified "2023-10-16" @default.
- W4214967508 title "An inhomogeneous singular perturbation problem for the $p(x)-$Laplacian" @default.
- W4214967508 doi "https://doi.org/10.48550/arxiv.1510.00316" @default.
- W4214967508 hasPublicationYear "2015" @default.
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