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.
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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)
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The research for this paper was done in part while on sabbatical at the Courant Institute of Mathematical Sciences, New York University.
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Williams, S.A. A partial solution of the Pompeiu problem. Math. Ann. 223, 183–190 (1976). https://doi.org/10.1007/BF01360881
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DOI: https://doi.org/10.1007/BF01360881