A new result on the Pompeiu problem

Author:
R. Dalmasso

Journal:
Trans. Amer. Math. Soc. **352** (2000), 2723-2736

MSC (1991):
Primary 35N05

DOI:
https://doi.org/10.1090/S0002-9947-99-02533-7

Published electronically:
October 15, 1999

MathSciNet review:
1694284

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Abstract | References | Similar Articles | Additional Information

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 LMC-IMAG, Equipe EDP, BP 53, F-38041 Grenoble Cedex 9, France

Email:
robert.dalmasso@imag.fr

DOI:
https://doi.org/10.1090/S0002-9947-99-02533-7

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