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A new result on the Pompeiu problem

Author: R. Dalmasso
Journal: Trans. Amer. Math. Soc. 352 (2000), 2723-2736
MSC (1991): Primary 35N05
Published electronically: October 15, 1999
MathSciNet review: 1694284
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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

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

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