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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
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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  • [BB] R. Bañuelos and K. Burdzy, On the ``hot spots'' conjecture of J. Rauch, J. Funct. Anal. 164 (1999), 1-33. CMP 99:13
  • [BaB] R. Bass and K. Burdzy, Fiber Brownian motion and the ``hot spots'' problem, preprint.
  • [Bl] T. Bulfinch, Bulfinch's mythology, Harper Collins, New York, 1991.
  • [BW] K. Burdzy and W. Werner, A counterexample to the ``hot spots'' conjecture., Annals of Math. 149 (1999), 309-317. MR 2000b:35044
  • [CH] R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. I, John Wiley & Sons, New York, 1953. MR 16:426a
  • [F] L. Friedlander, An inequality between Dirichlet and Neumann eigenvalues in a centrally symmetric domain, personal communication.
  • [JK] D. Jerison and C. Kenig, The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc. 4 (1981), 203-207. MR 84a:35064
  • [K] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, vol. 33, Springer-Verlag, Berlin, 1985, Lecture Notes in Mathematics 1150. MR 87a:35001
  • [L] H. Lewy, On conjugate solutions of certain partial differential equations, Comm. Pure Appl. Math. 33 (1980), 441-445. MR 81j:35036
  • [M] A. D. Melas, On the nodal line of the second eigenfunction of the Laplacian in $\mathbf R^{2}$, J. Diff. Geometry 35 (1992), 255-263. MR 93g:35100
  • [N] N. Nadirashvili, Multiple eigenvalues of the Laplace operator, Math. USSR Sbornik 61 (1988), 225-238. MR 89a:58113
  • [P] L. Payne, Isoperimetric inequalities and their applications, S.I.A.M. Rev. 9 (1967), 453-488. MR 36:2058
  • [Pl] A. Pleijel, Remarks on Courant's nodal line theorem, Comm. Pure Appl. Math. 9 (1956), 543-550. MR 18:315d
  • [PW] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984. MR 86f:35034
  • [R] J. Rauch, Five problems: an introduction to the qualitative theory of partial differential equations, Partial Differential Equations and Related Topics (Jerome A. Goldstein, ed.), Springer-Verlag, Berlin, 1974, pp. 355-369, Lecture Notes in Mathematics 446. MR 58:22963
  • [Z] D. Zanger, Thesis MIT, June 1997.

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Additional Information

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

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

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

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