Symmetry problem

Ramm, A.

doi:10.1090/S0002-9939-2012-11400-5

Symmetry problem
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by A. G. Ramm PDF

Proc. Amer. Math. Soc. 141 (2013), 515-521 Request permission

Abstract:

A novel approach to an old symmetry problem is developed. A new proof is given for the following symmetry problem, studied earlier: if $\Delta u=1$ in $D\subset \mathbb {R}^3$, $u=0$ on $S$, the boundary of $D$, and $u_N=const$ on $S$, then $S$ is a sphere. It is assumed that $S$ is a Lipschitz surface homeomorphic to a sphere. This result has been proved in different ways by various authors. Our proof is based on a simple new idea.

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