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.

**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 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

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