Geodesic
The “hot spots” conjecture for domains with two axes of symmetry
Jerison, David 1   ; Nadirashvili, Nikolai 2
1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
2 Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Journal of the American Mathematical Society, Tome 13 (2000) no. 4, pp. 741-772 Cet article a éte moissonné depuis la source American Mathematical Society

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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.
DOI : 10.1090/S0894-0347-00-00346-5

Jerison, David  1   ; Nadirashvili, Nikolai  2

1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
2 Department of Mathematics, University of Chicago, Chicago, Illinois 60637 
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Jerison, David; Nadirashvili, Nikolai. The “hot spots” conjecture for domains with two axes of symmetry. Journal of the American Mathematical Society, Tome 13 (2000) no. 4, pp. 741-772. doi: 10.1090/S0894-0347-00-00346-5

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