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


Authors: Nicola Garofalo and Elena Sartori
Journal: Proc. Amer. Math. Soc. 129 (2001), 3603-3610
MSC (1991): Primary 35K55
DOI: https://doi.org/10.1090/S0002-9939-01-05993-7
Published electronically: June 28, 2001
MathSciNet review: 1860493
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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 $a$ modeled on $\vert Du\vert^{p-2}$. We consider a solution satisfying the boundary conditions: $u(x,t)=f(t)$ for $(x,t)\in \partial \Omega_{1} \times (O,T)$, and $u(x,0)=0$, $u\rightarrow 0$ 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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Additional Information

Nicola Garofalo
Affiliation: Institut Mittag-Leffler, Auravägen 17, S-182 62 Djursholm, Sweden
Address at time of publication: Department of Mathematics, The Johns Hopkins University, 3400 N. Charles St., Baltimore, Maryland 21218
Email: garofalo@ml.kva.se

Elena Sartori
Affiliation: Dipartimento di Metodi e Modelli Matematici, Universitá di Padova, 35131 Padova, Italy
Email: sartori@math.unipd.it

DOI: https://doi.org/10.1090/S0002-9939-01-05993-7
Received by editor(s): April 18, 2000
Published electronically: June 28, 2001
Additional Notes: The first author was supported by NSF Grant No. DMS-9706892.
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2001 American Mathematical Society

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