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Symmetry in a free boundary problem for degenerate parabolic equations on unbounded domains

Authors:	Nicola Garofalo Elena Sartori

Institution:	Institut Mittag-Leffler, Auravägen 17, S-182 62 Djursholm, Sweden ; Dipartimento di Metodi e Modelli Matematici, Universitá di Padova, 35131 Padova, Italy

Abstract:	We use the method of Alexandroff-Serrin to establish the spherical symmetry of the ground domain and of the weak solution to a free boundary problem for a class of quasi-linear parabolic equations in an unbounded cylinder $\Omega \times (0,T)$ , where $\Omega = (\mathbb{R} ^{n} \backslash \overline{\Omega_{1}})$ , with $\Omega_{1}\subset \mathbb R^n$ a simply connected bounded domain. The equations considered are of the type $u_{t} - div (a(u,\vert Du\vert)Du) = c(u,\vert Du\vert)$ , with modeled on $\vert Du\vert^{p-2}$ . We consider a solution satisfying the boundary conditions: for $(x,t)\in \partial \Omega_{1} \times (O,T)$ , and , $u\rightarrow 0$ as $\vert x\vert\rightarrow\infty$ . We show that the overdetermined co-normal condition $a(u,\vert Du\vert)\frac{\partial u}{\partial\nu}=g(t)$ for $(x,t)\in \partial \Omega_{1} \times (O,T)$ , with $g(\overline T) > 0$ for at least one value $\overline T \in (0,T)$ , forces the spherical symmetry of the ground domain and of the solution.

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