Symmetry problem
Author:
A. G. Ramm
Journal:
Proc. Amer. Math. Soc. 141 (2013), 515521
MSC (2010):
Primary 35J05, 31B20
Published electronically:
May 31, 2012
MathSciNet review:
2996955
Fulltext PDF
Abstract 
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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 in , on , the boundary of , and on , then is a sphere. It is assumed that 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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F. Weinberger, Remark on the preceding paper of Serrin, Arch.
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 A. D. Alexandrov, A characteristic property of a sphere, Ann. di Matem., 58 (1962), 303315. MR 0143162 (26:722)
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 T. Chatelain and A. Henrot, Some results about Schiffer's conjectures, Inverse Problems, 15 (1999), 647658. MR 1696934 (2000e:35019)
 5.
 N. S. Hoang and A. G. Ramm, Symmetry problems. II, Annal. Polon. Math., 96, N1 (2009), 6164. MR 2506593 (2010f:35046)
 6.
 G. Korn and T. Korn, Mathematical Handbook for Scientists and Engineers, McGrawHill, New York, 1968. MR 0220560 (36:3618)
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 A. A. Kosmodem'yanskiĭ, A converse of the mean value theorem for harmonic functions, Russ. Math. Surveys, 36, N5 (1981), 159160. MR 637445 (84d:31001)
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 A. G. Ramm, Scattering by obstacles, D. Reidel, Dordrecht, 1986. MR 847716 (87k:35197)
 9.
 A. G. Ramm, The Pompeiu problem, Applicable Analysis, 64, N12 (1997), 1926. MR 1460069 (98d:35036)
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 A. G. Ramm, Necessary and sufficient condition for a domain, which fails to have Pompeiu property, to be a ball, Journ. of Inverse and IllPosed Probl., 6, N2 (1998), 165171. MR 1637368 (99f:35026)
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 A. G. Ramm, A symmetry problem, Ann. Polon. Math., 92 (2007), 4954. MR 2318510 (2008d:31003)
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 J. Serrin, A symmetry problem in potential theory, Arch. Rat. Mech. Anal., 43 (1971), 304318. MR 0333220 (48:11545)
 15.
 H. Weinberger, Remark on the preceding paper of Serrin, Arch. Rat. Mech. Anal., 43 (1971), 319320. MR 0333221 (48:11546)
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Additional Information
A. G. Ramm
Affiliation:
Department of Mathematics, Kansas State University, Manhattan, Kansas 665062602
Email:
ramm@math.ksu.edu
DOI:
http://dx.doi.org/10.1090/S000299392012114005
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.
