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# The Free Boundary of a Semilinear Elliptic Equation

Avner Friedman and Daniel Phillips
Transactions of the American Mathematical Society
Vol. 282, No. 1 (Mar., 1984), pp. 153-182
DOI: 10.2307/1999583
Stable URL: http://www.jstor.org/stable/1999583
Page Count: 30
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## Abstract

The Dirichlet problem $\Delta u = \lambda f(u)$ in a domain $\Omega, u = 1$ on $\partial\Omega$ is considered with $f(t) = 0$ if $t \leqslant 0, f(t) > 0$ 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 $\lambda \rightarrow \infty$. 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 $\partial\Omega$

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