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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The free boundary of a semilinear elliptic equation


Authors: Avner Friedman and Daniel Phillips
Journal: Trans. Amer. Math. Soc. 282 (1984), 153-182
MSC: Primary 35J65; Secondary 35J85, 35R35
DOI: https://doi.org/10.1090/S0002-9947-1984-0728708-4
MathSciNet review: 728708
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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 \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 $.


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DOI: https://doi.org/10.1090/S0002-9947-1984-0728708-4
Keywords: Semilinear elliptic equation, free boundary, variational inequality, maximal and minimal solutions, minimizer, asymptotic behavior, reduced boundary, nonnegative mean curvature
Article copyright: © Copyright 1984 American Mathematical Society