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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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The free boundary of a semilinear elliptic equation
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by Avner Friedman and Daniel Phillips PDF
Trans. Amer. Math. Soc. 282 (1984), 153-182 Request permission

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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Additional Information
  • © Copyright 1984 American Mathematical Society
  • 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