## Symmetry problem

HTML articles powered by AMS MathViewer

- by A. G. Ramm PDF
- Proc. Amer. Math. Soc.
**141**(2013), 515-521 Request permission

## Abstract:

A novel approach to an old symmetry problem is developed. A new proof is given for the following symmetry problem, studied earlier: if $\Delta u=1$ in $D\subset \mathbb {R}^3$, $u=0$ on $S$, the boundary of $D$, and $u_N=const$ on $S$, then $S$ is a sphere. It is assumed that $S$ is a Lipschitz surface homeomorphic to a sphere. This result has been proved in different ways by various authors. Our proof is based on a simple new idea.## References

- A. D. Alexandrov,
*A characteristic property of spheres*, Ann. Mat. Pura Appl. (4)**58**(1962), 303–315. MR**143162**, DOI 10.1007/BF02413056 - Tewodros Amdeberhan,
*Two symmetry problems in potential theory*, Electron. J. Differential Equations (2001), No. 43, 5. MR**1836811** - Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi,
*Higher transcendental functions. Vols. I, II*, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR**0058756** - Thierry Chatelain and Antoine Henrot,
*Some results about Schiffer’s conjectures*, Inverse Problems**15**(1999), no. 3, 647–658. MR**1696934**, DOI 10.1088/0266-5611/15/3/301 - N. S. Hoang and A. G. Ramm,
*Symmetry problems. II*, Ann. Polon. Math.**96**(2009), no. 1, 61–64. MR**2506593**, DOI 10.4064/ap96-1-5 - Granino A. Korn and Theresa M. Korn,
*Mathematical handbook for scientists and engineers*, Second, enlarged and revised edition, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1968. MR**0220560** - A. A. Kosmodem′yanskiĭ Jr.,
*Converse of the mean value theorem for harmonic functions*, Uspekhi Mat. Nauk**36**(1981), no. 5(221), 175–176 (Russian). MR**637445** - A. G. Ramm,
*Scattering by obstacles*, Mathematics and its Applications, vol. 21, D. Reidel Publishing Co., Dordrecht, 1986. MR**847716**, DOI 10.1007/978-94-009-4544-9 - A. G. Ramm,
*The Pompeiu problem*, Appl. Anal.**64**(1997), no. 1-2, 19–26. MR**1460069**, DOI 10.1080/00036819708840520 - A. G. Ramm,
*Necessary and sufficient condition for a domain, which fails to have Pompeiu property, to be a ball*, J. Inverse Ill-Posed Probl.**6**(1998), no. 2, 165–171. MR**1637368**, DOI 10.1515/jiip.1998.6.2.165 - Alexander G. Ramm,
*Inverse problems*, Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York, 2005. Mathematical and analytical techniques with applications to engineering; With a foreword by Alan Jeffrey. MR**2838778** - A. G. Ramm,
*A symmetry problem*, Ann. Polon. Math.**92**(2007), no. 1, 49–54. MR**2318510**, DOI 10.4064/ap92-1-5 - A. G. Ramm and E. I. Shifrin,
*The problem of symmetry in a problem of the theory of elasticity on a plane crack of normal discontinuity*, Prikl. Mat. Mekh.**69**(2005), no. 1, 135–143 (Russian, with Russian summary); English transl., J. Appl. Math. Mech.**69**(2005), no. 1, 127–134. MR**2158714**, DOI 10.1016/j.jappmathmech.2005.01.012 - James Serrin,
*A symmetry problem in potential theory*, Arch. Rational Mech. Anal.**43**(1971), 304–318. MR**333220**, DOI 10.1007/BF00250468 - H. F. Weinberger,
*Remark on the preceding paper of Serrin*, Arch. Rational Mech. Anal.**43**(1971), 319–320. MR**333221**, DOI 10.1007/BF00250469

## Additional Information

**A. G. Ramm**- Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506-2602
- Email: ramm@math.ksu.edu
- Received by editor(s): December 6, 2010
- Received by editor(s) in revised form: June 25, 2011
- Published electronically: May 31, 2012
- Communicated by: Matthew J. Gursky
- © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**141**(2013), 515-521 - MSC (2010): Primary 35J05, 31B20
- DOI: https://doi.org/10.1090/S0002-9939-2012-11400-5
- MathSciNet review: 2996955