A new result on the Pompeiu problem
Author:
R. Dalmasso
Journal:
Trans. Amer. Math. Soc. 352 (2000), 27232736
MSC (1991):
Primary 35N05
Published electronically:
October 15, 1999
MathSciNet review:
1694284
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Abstract: A nonempty bounded open set () is said to have the Pompeiu property if and only if the only continuous function on for which the integral of over is zero for all rigid motions of is . We consider a nonempty bounded open set with Lipschitz boundary and we assume that the complement of is connected. We show that the failure of the Pompeiu property for implies some geometric conditions. Using these conditions we prove that a special kind of solid tori in , , has the Pompeiu property. So far the result was proved only for solid tori in . We also examine the case of planar domains. Finally we extend the example of solid tori to domains in bounded by hypersurfaces of revolution.
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Additional Information
R. Dalmasso
Affiliation:
Laboratoire LMCIMAG, Equipe EDP, BP 53, F38041 Grenoble Cedex 9, France
Email:
robert.dalmasso@imag.fr
DOI:
http://dx.doi.org/10.1090/S0002994799025337
PII:
S 00029947(99)025337
Keywords:
Pompeiu problem,
Schiffer's conjecture
Received by editor(s):
June 6, 1997
Received by editor(s) in revised form:
January 14, 1998
Published electronically:
October 15, 1999
Article copyright:
© Copyright 2000 American Mathematical Society
