Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A symmetry theorem revisited

Authors: John Lewis and Andrew Vogel
Journal: Proc. Amer. Math. Soc. 130 (2002), 443-451
MSC (1991): Primary 31B05, 31B20
Published electronically: June 6, 2001
MathSciNet review: 1862124
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information


We show that if harmonic measure and Hausdorff measure are equal on the boundary of certain domains in Euclidean $n$-space, then these domains are necessarily balls.

References [Enhancements On Off] (What's this?)

  • [Hel69] L. L. Helms, Introduction to potential theory, Pure and Applied Mathematics, Vol. XXII, Wiley-Interscience A Division of John Wiley & Sons, New York-London-Sydney, 1969. MR 0261018
  • [KL37] M. Keldysh and M. Lavrentiev, Sur la représentation conforme des domain limités par des courbes rectifiables, Ann. Sci. École Norm. Sup. 54 (1937), 1-38.
  • [LV] John L. Lewis and Andrew L. Vogel, On pseudospheres that are quasispheres, Rev. Mat. Iberoamericana, To appear.
  • [LV91] John L. Lewis and Andrew Vogel, On pseudospheres, Rev. Mat. Iberoamericana 7 (1991), no. 1, 25–54. MR 1109479, 10.4171/RMI/104
  • [LV92] John L. Lewis and Andrew Vogel, On some almost everywhere symmetry theorems, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989) Progr. Nonlinear Differential Equations Appl., vol. 7, Birkhäuser Boston, Boston, MA, 1992, pp. 347–374. MR 1167849
  • [Mat95] Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890
  • [MMV96] Pertti Mattila, Mark S. Melnikov, and Joan Verdera, The Cauchy integral, analytic capacity, and uniform rectifiability, Ann. of Math. (2) 144 (1996), no. 1, 127–136. MR 1405945, 10.2307/2118585

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 31B05, 31B20

Retrieve articles in all journals with MSC (1991): 31B05, 31B20

Additional Information

John Lewis
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027

Andrew Vogel
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244

Keywords: Harmonic measure, Hausdorff measure, quasiconformal, Green's function, Dirichlet problem
Received by editor(s): June 20, 2000
Published electronically: June 6, 2001
Additional Notes: The first author was supported in part by an NSF grant
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2001 American Mathematical Society