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Transactions of the American Mathematical Society

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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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  • 1. C. A. Berenstein, An inverse spectral theorem and its relation to the Pompeiu problem, J. Anal. Math. 37 (1980), 128-144. MR 82b:35031
  • 2. C. A. Berenstein and D. Khavinson, Do solid tori have the Pompeiu property?, Expo. Math. 15 (1997), 87-93. CMP 97:09
  • 3. L. Brown, B. M. Schreiber and B. A. Taylor, Spectral synthesis and the Pompeiu problem, Ann. Inst. Fourier 23 (1973), 125-154. MR 50:4979
  • 4. P. Ebenfelt, Singularities of solutions to a certain Cauchy problem and an application to the Pompeiu problem, Duke Math. J. 71 (1993), 119-142. MR 94k:35006
  • 5. P. Ebenfelt, Some results on the Pompeiu problem, Ann. Acad. Sci. Fennicae 18 (1993), 323-341. MR 95b:30061
  • 6. P. Ebenfelt, Propagation of singularities from singular and infinite points in certain analytic Cauchy problems and an application to the Pompeiu problem, Duke Math. J. 73 (1994), 561-582. MR 95c:35012
  • 7. H. Flanders, A proof of Minkowski's inequality for convex curves, Amer. Math. Monthly 75 (1968), 581-593. MR 38:1609
  • 8. N. Garofalo and F. Segala, New results on the Pompeiu problem, Trans. Amer. Math. Soc. 325-1 (1991), 273-286. MR 91h:35322
  • 9. N. Garofalo and F. Segala, Another step toward the solution of the Pompeiu problem in the plane, Commun. in Partial Differential Equations 18 (1993), 491-503. MR 94j:30039
  • 10. N. Garofalo and F. Segala, Univalent functions and the Pompeiu problem, Trans. Amer. Math. Soc. 346 (1994), 137-146. MR 95b:30062
  • 11. G. Johnsson, The Cauchy problem in $\mathbb C^{n}$ for linear second order partial differential equations with data on a quadric surface, Trans. Amer. Math. Soc. 344 (1994), 1-48. MR 94m:35008
  • 12. P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J. 35 (1986), 681-703. MR 88b:35072
  • 13. S. A. Williams, A partial solution to the Pompeiu problem, Math. Ann. 223-2 (1976), 183-190. MR 54:2996
  • 14. S. A. Williams, Analycity of the boundary for Lipschitz domains without the Pompeiu property, Indiana Univ. Math. J. 30 (1981), 357-369. MR 82j:31009
  • 15. S. T. Yau, Problem Section, in Seminar on Differential Geometry, edited by S. T. Yau, Annals of Math. Studies, Princeton, N.J., 1982. MR 83e:53029
  • 16. L. Zalcman, Offbeat integral geometry, Amer. Math. Monthly 87 (1980), 161-175. MR 81b:53046
  • 17. L. Zalcman, A bibliographic survey of the Pompeiu problem, in Approximation by solutions of partial differential equations, B. Fuglede et al. (eds.), Kluwer Acad. Publ., 1992, 185-194. MR 93e:26001

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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