The free boundary of a semilinear elliptic equation

The Dirichlet problem $\Delta u = \lambda \,f(u)$ in a domain $\Omega ,\,u = 1$ on $\partial\Omega$ is considered with $f(t) = 0$ if $t \leq 0,\,f(t) > 0$ if $t > 0,\,f(t) \sim {t^p}$ if $t \downarrow 0,0 < p < 1;f(t)$ is not monotone in general. The set $\{ u = 0\}$ and the ``free boundary'' $\partial \{ u = 0\}$ are studied. Sharp asymptotic estimates are established as $\lambda \to \infty$. For suitable $f$, under the assumption that $\Omega$ is a two-dimensional convex domain, it is shown that $\{ u = 0\}$ is a convex set. Analogous results are established also in the case where $\partial u/\partial v + \mu (u - 1) = 0$ on $\partial \Omega$.