The ``hot spots'' conjecture for domains with two axes of symmetry

Authors:
David Jerison and Nikolai Nadirashvili

Journal:
J. Amer. Math. Soc. **13** (2000), 741-772

MSC (1991):
Primary 35J25, 35B65; Secondary 35J05

DOI:
https://doi.org/10.1090/S0894-0347-00-00346-5

Published electronically:
July 21, 2000

MathSciNet review:
1775736

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Consider a Neumann eigenfunction with lowest nonzero eigenvalue of a convex planar domain with two axes of symmetry. We show that the maximum and minimum of the eigenfunction are achieved at points on the boundary only. We deduce J. Rauch's ``hot spots'' conjecture: if the initial temperature distribution is not orthogonal to the first nonzero eigenspace, then the point at which the temperature achieves its maximum tends to the boundary. This was already proved by Bañuelos and Burdzy in the case in which the eigenspace is one dimensional. We introduce here a new technique based on deformations of the domain that applies to the case of multiple eigenvalues.

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Additional Information

**David Jerison**

Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Email:
jerison@math.mit.edu

**Nikolai Nadirashvili**

Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637

Email:
nicholas@math.uchicago.edu

DOI:
https://doi.org/10.1090/S0894-0347-00-00346-5

Keywords:
Convex domains,
eigenfunctions

Received by editor(s):
August 6, 1999

Received by editor(s) in revised form:
October 13, 1999

Published electronically:
July 21, 2000

Additional Notes:
The first author was partially supported by NSF grants DMS-9401355 and DMS-9705825. The second author was partially supported by NSF grant DMS-9971932.

Article copyright:
© Copyright 2000
American Mathematical Society