Abstract
A nonempty bounded open subset D of ℝn is said to have the Pompeiu property if and only if for every continuous complex-valued function f on ℝn which does not vanish identically there is a rigid motion σ of ℝn onto itself — taking D onto σ(D) — such that the integral of f over σ(D) is not zero. This article gives a partial solution of the Pompeiu problem, the problem of finding all sets D with the Pompeiu property.
In the special case that D is the interior of a homeomorphic image of an(n−1)-dimensional sphere, the main result states that if D has a portion of an(n−1)-dimensional real analytic surface on its boundary, then either D has the Pompeiu property or any connected real analytic extension of the surface also lies on the boundary of D. Thus, for example, any such region D having a portion of a hyperplane as part of its boundary must have the Pompeiu property, since the entire hyperplane cannot lie in the boundary of the bounded set D.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Brown, L., Schreiber, B., Taylor, B. A.: Spectral synthesis and the Pompeiu problem. Ann. Inst. Fourier, Grenoble,23, 125–154 (1973)
Courant, R., Hilbert, D.: Methods of mathematical physics. Vol. I. New York: Interscience 1953
Hörmander, L.: Linear partial differential operators. Berlin, Heidelberg, New York: Springer 1969
Gunning, R., Rossi, H.: Analytic functions of several complex variables. Englewood Cliffs, N. J.: Prentice-Hall 1965
Trèves, F.: Linear partial differential equations with constant coefficients. New York: Gordon and Breach 1966
Dunford, N., Schwartz, J.: Linear operators. Vol. II. New York: Interscience 1963
Morrey, C. B., Jr.: On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations. Part I. Amer. J. of Math.80, 198–218 (1958)
Spanier, E.: Algebraic topology. New York: McGraw-Hill 1966
Courant, R., Hilbert, D.: Methods of mathematical physics. Vol. II. New York: Interscience 1962
Serrin, J.: A symmetry problem in potential theory. Arch. Rat. Mech. Anal.43, 304–318 (1971)
Author information
Authors and Affiliations
Additional information
The research for this paper was done in part while on sabbatical at the Courant Institute of Mathematical Sciences, New York University.
Rights and permissions
About this article
Cite this article
Williams, S.A. A partial solution of the Pompeiu problem. Math. Ann. 223, 183–190 (1976). https://doi.org/10.1007/BF01360881
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01360881