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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
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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