Viscosity solutions and regularity of the free boundary for the limit of a two phase singular perturbation problem

LEDERMAN, CLAUDIA; WOLANSKI, NOEMÍ

ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA CL. DI SCIENZE - IV

Scuola Normale Superiore

Lugar: Pisa; Año: 1998 vol. 27 p. 253 - 288

0391-173X

In this paper we are concerned with the following problem: Study the limit as $\ep\to0$, of solutions $\uep(x)$ to the equation: $$ \Delta\uep=\beta_\ep(\uep) \tag $E_\ep$ $$ where $\ep>0$ and $\beta_\ep(s)=\frac1\ep\beta(\frac s\ep)$. Here $\beta$ is a Lipschitz continuous function with $\beta>0$ in (0,1) and $\beta\equiv0$ outside (0,1) and $\int \beta(s)ds=M$. We consider a family $\uep$ of uniformly bounded solutions to $E_\ep$ in a domain $\Omega\subset \Bbb R^N$ and we prove that, under suitable assumptions, the limit function $u$ is a solution to $$ \aligned \Delta u&=0\qquad\qquad\qquad\qquad\qquad\,\qquad\text{in }\Omega\setminus\partial\{u>0\}\\ u&=0,\ \ (u^+_\nu)^2-(u^-_\nu)^2=2M\qquad\text{on }\Omega\cap\fb \endaligned \tag $E$ $$ in a pointwise sense at ``regular'' free boundary points and in a viscosity sense. Then, we prove the regularity of the free boundary. In fact, we prove that in the absence of zero phase, if $u^-$ is nondegenerate at $x_0\in \Omega\cap\fb$, then the free boundary is a $C^{1,\a}$ surface in a neighborhood of $x_0$. Therefore, $u$ is a classical solution to $(E)$ in that neighborhood. For the general two phase case (which includes, in particular, the one phase case) we prove that, under nondegeneracy assumptions on $u$, if the free boundary has an inward unit normal in the measure theoretic sense at a point $x_0\in \Omega\cap\fb$, then the free boundary is a $C^{1,\a}$ surface in a neighborhood of $x_0$.