A singular perturbation problem for a quasilinear operator satisfying the natural growth conditions of Lieberman

NOEMI WOLANSKI; MARTÍNEZ, SANDRA RITA

Estocolmo, Suecia

Workshop; Free Boundary Problems. Theory and Applications; 2008

In this paper we study the following problem. For   $ep>0$, $u^ep$ a solution of $$ L u^{ep}:= div,(g(| abla uep|)}{| nabla u^ep|} nabla uep)=beta_{ep}(u^{ep}), u^{ep}geq 0.$$ Here $beta_{ep}(s)= rac{1}{ep}eta(s/ep), $ with  $beta$ Lipschitz $beta>0$ in $(0,1)$ and $beta=0$ otherwise.We are interested in  the limiting problem, when $ep o 0$. As inprevious work with $L=Delta$ or $L=Delta_p$ we prove, under appropriate assumptions, that any limiting function is a weak solution to a free boundary problem. Moreover, for nondegenerate limits we prove that the reduced free boundary is a $C^{1,alpha}$ surface. This result is new even for $Delta_p$. Throughout the paper, we  assume that $g$ satisfies the conditions introduced by G. Lieberman in {The natural generalization of the natural conditions of Ladyzhenskaya and Ural´tseva for elliptic equations}, Comm. Partial Differential Equations {16} (1991), no. 2-3, 311--361.