## A new result on the Pompeiu problem

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- by R. Dalmasso PDF
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**352**(2000), 2723-2736 Request permission

## 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
- Received by editor(s): June 6, 1997
- Received by editor(s) in revised form: January 14, 1998
- Published electronically: October 15, 1999
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**352**(2000), 2723-2736 - MSC (1991): Primary 35N05
- DOI: https://doi.org/10.1090/S0002-9947-99-02533-7
- MathSciNet review: 1694284