A symmetry problem from probability

Authors:
Stephen J. Fromm and Patrick McDonald

Journal:
Proc. Amer. Math. Soc. **125** (1997), 3293-3297

MSC (1991):
Primary 35J40, 60J65, 58G32

DOI:
https://doi.org/10.1090/S0002-9939-97-04162-2

MathSciNet review:
1425121

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Abstract | References | Similar Articles | Additional Information

Abstract: We examine solutions of two related boundary value problems for smooth domains in Euclidean space which arise from variational problems in probability. We show that the existence of solutions to each problem implies that the domain is a sphere.

**[B]**A. Bennett,*Symmetry in an overdetermined fourth order elliptic boundary value problem*, SIAM J. Math. Anal.**17**(1986), 1354-1358. MR**87i:35059****[KM1]**K. K. J. Kinateder and P. McDonald,*Brownian functionals on hypersurfaces in Euclidean space*, Proc. Amer. Math. Soc.**125**(1997), 1815-1822. CMP**97:07****[KM2]**K. K. J. Kinateder and P. McDonald,*Hypersurfaces in and the variance of exit times for Brownian motion*, Proc. Amer. Math. Soc. (to appear). CMP**96:16****[S]**J. Serrin,*A symmetry problem in potential theory*, Arch. Rational Mech. Anal.**43**(1971), 304-318. MR**48:11545****[W]**H. F. Weinberger,*Remark on the preceding paper of Serrin*, Arch. Rational Mech. Anal.**43**(1971), 319-320. MR**48:11546**

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Additional Information

**Stephen J. Fromm**

Affiliation:
Department of Mathematics, University of Wyoming, Laramie, Wyoming 82071

Email:
fromm@uwyo.edu

**Patrick McDonald**

Affiliation:
Department of Mathematics, The Ohio State University, Columbus, Ohio 43210

Address at time of publication:
Department of Mathematics, New College, University of South Florida, Sarasota, Florida 34243

Email:
pmacdona@virtu.sar.usf.edu

DOI:
https://doi.org/10.1090/S0002-9939-97-04162-2

Keywords:
Poisson problem,
overdetermined boundary value problem

Received by editor(s):
June 5, 1996

Communicated by:
Jeffrey Rauch

Article copyright:
© Copyright 1997
American Mathematical Society