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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Symmetry problem


Author: A. G. Ramm
Journal: Proc. Amer. Math. Soc. 141 (2013), 515-521
MSC (2010): Primary 35J05, 31B20
Published electronically: May 31, 2012
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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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Additional Information

A. G. Ramm
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506-2602
Email: ramm@math.ksu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11400-5
PII: S 0002-9939(2012)11400-5
Keywords: Symmetry problems, potential theory.
Received by editor(s): December 6, 2010
Received by editor(s) in revised form: June 25, 2011
Published electronically: May 31, 2012
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.