ISSN:
1432-0673
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Abstract. In this paper, we extend a classical result by J. Serrin [15] to exterior domains $\rit ^n \setminus \overline{\Omega}$ , where Ω is a bounded domain. We prove, under some hypotheses on f, that if there exists a solution of $\Delta u +f(u) =0$ in $\rit ^n \setminus \overline{\Omega}$ satisfying the overdetermined boundary conditions that ${\partial u} /{\partial\nu}$ and u are constant on $\partial \Omega$ , and such that $0\leq u \leq u \mid_{\partial \Omega}$ , then the domain Ω is a ball. Under different assumptions on f, this result has been obtained by W. Reichel in [13]. The main result here covers new cases like $f(u)=u^p$ with ${n \over {n-2}} 〈p\leq {{n+2}\over{n-2}}$ . When Ω is a ball, almost the same proof allows us to derive the symmetry of positive bounded solutions satisfying only the Dirichlet condition that u is constant on $\partial \Omega$ . Our method relies on Kelvin transforms, various forms of the maximum principle and the device of moving planes up to a critical position.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s002050050103
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