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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A nonempty bounded open set $\Omega \subset {\mathbb {R}}^{n}$ ($n \geq 2$) is said to have the *Pompeiu property* if and only if the only continuous function $f$ on ${\mathbb {R}}^{n}$ for which the integral of $f$ over $\sigma (\Omega )$ is zero for all rigid motions $\sigma$ of ${\mathbb {R}}^{n}$ is $f \equiv 0$. We consider a nonempty bounded open set $\Omega \subset {\mathbb {R}}^{n}$ $(n \geq 2)$ with Lipschitz boundary and we assume that the complement of $\overline {\Omega }$ is connected. We show that the failure of the Pompeiu property for $\Omega$ implies some geometric conditions. Using these conditions we prove that a special kind of solid tori in ${\mathbb {R}}^{n}$, $n \geq 3$, has the Pompeiu property. So far the result was proved only for solid tori in ${\mathbb {R}}^{4}$. We also examine the case of planar domains. Finally we extend the example of solid tori to domains in ${\mathbb {R}}^{n}$ 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

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