Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


New results on the Pompeiu problem
HTML articles powered by AMS MathViewer

by Nicola Garofalo and Fausto Segàla PDF
Trans. Amer. Math. Soc. 325 (1991), 273-286 Request permission


Let ${p_N}(w) = \sum \nolimits _{k = 0}^N {{a_k}{w^k}}$, $w \in \mathbb {C}$, $N \in \mathbb {N}$, be a polynomial with complex coefficients. In this paper we prove that if $D \subset {\mathbb {R}^2}$ is a simply-connected bounded open set whose boundary is a closed, simple curve parametrized by $x(s) = {x_1}(s) + i{x_2}(s) = {p_N}({e^{is}})$, $s \in [ - \pi ,\pi ]$, then $D$ has the Pompeiu property unless $N = 1$ and ${p_1}(w) = {a_1}w + {a_2}$ in which case $D$ is a disk. This result supports the conjecture that modulo sets of zero two-dimensional Lebesgue measure, the disk is the only simply-connected, bounded open set which fails to have the Pompeiu property.
  • Carlos Alberto Berenstein, An inverse spectral theorem and its relation to the Pompeiu problem, J. Analyse Math. 37 (1980), 128–144. MR 583635, DOI 10.1007/BF02797683
  • Leon Brown and Jean-Pierre Kahane, A note on the Pompeiu problem for convex domains, Math. Ann. 259 (1982), no. 1, 107–110. MR 656655, DOI 10.1007/BF01456832
  • Leon Brown, Bertram M. Schreiber, and B. Alan Taylor, Spectral synthesis and the Pompeiu problem, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3, 125–154 (English, with French summary). MR 352492
  • Luis A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (1977), no. 3-4, 155–184. MR 454350, DOI 10.1007/BF02392236
  • L. Tchakaloff, Sur un problème de D. Pompéiu, Annuaire [Godišnik] Univ. Sofia. Fac. Phys.-Math. Livre 1. 40 (1944), 1–14 (Bulgarian, with French summary). MR 0031980
  • D. Pompeiu, Sur certains systèmes d’équations linéaires et sur une propriété intégrale des fonctions de plusieurs variables, C. R. Acad. Sci. Paris 188 (1929), 1.138-1.139. —, Sur une propriété intégrales des fonctions de deux variables réelles, Bull. Sci. Acad. Royale Belgique 15 (1929), 265-269. B. Riemann, Sullo svolgimento del quoziente di due serie ipergeometriche in funzione continua infinita, Complete Works, Dover, New York, 1953.
  • Stephen A. Williams, A partial solution of the Pompeiu problem, Math. Ann. 223 (1976), no. 2, 183–190. MR 414904, DOI 10.1007/BF01360881
  • Stephen A. Williams, Analyticity of the boundary for Lipschitz domains without the Pompeiu property, Indiana Univ. Math. J. 30 (1981), no. 3, 357–369. MR 611225, DOI 10.1512/iumj.1981.30.30028
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 35R30, 31B20, 35J05
  • Retrieve articles in all journals with MSC: 35R30, 31B20, 35J05
Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 325 (1991), 273-286
  • MSC: Primary 35R30; Secondary 31B20, 35J05
  • DOI:
  • MathSciNet review: 994165