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The free boundary of a semilinear elliptic equation. (English) Zbl 0552.35079
This paper contains a detailed study of the Dirichlet problem \(\Delta u=\lambda f(u)\) in \(\Omega\), \((\lambda >0)\); \(u=1\) on \(\partial \Omega\), where f:\({\mathbb{R}}\to {\mathbb{R}}\) is a function satisfying \(f(t)=0\), if \(t\geq 0\), \(f(t)>0\), if \(t>0\), and \(f(t)\sim t^ p\) as \(t\downarrow 0\), \(0<p<1\). Special emphasis is layed on the study of the ”dead core” \(N_{\lambda}=\{u_{\lambda}=0\}\) and the properties of the ”free boundary” \(\partial N_{\lambda}\). For instance the authors prove that for \(\lambda\) large \(\partial N_{\lambda}\) is a smooth surface parallel to \(\partial \Omega\) at a distance \(\gamma /\sqrt{\lambda}+O(1/\lambda),\) \(\gamma\) constant; for convex domains \(\Omega \subset {\mathbb{R}}^ 2\) and for suitable f, \(N_{\lambda}\) is shown to be convex. Most of the results are extended to the case of the Rodin problem that is the boundary condition changes to \(\partial u/\partial \nu +\mu (u+1)=0\) on \(\partial \Omega\).
Reviewer: R.Landes

MSC:
35R35 Free boundary problems for PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
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