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On Neumann eigenfunctions in lip domains

Authors: Rami Atar and Krzysztof Burdzy
Journal: J. Amer. Math. Soc. 17 (2004), 243-265
MSC (2000): Primary 35J05; Secondary 60H30
Published electronically: February 11, 2004
MathSciNet review: 2051611
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Abstract: A “lip domain” is a planar set lying between graphs of two Lipschitz functions with constant 1. We show that the second Neumann eigenvalue is simple in every lip domain except the square. The corresponding eigenfunction attains its maximum and minimum at the boundary points at the extreme left and right. This settles the “hot spots” conjecture for lip domains as well as two conjectures of Jerison and Nadirashvili. Our techniques are probabilistic in nature and may have independent interest.

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Additional Information

Rami Atar
Affiliation: Department of Electrical Engineering, Technion–Israel Institute of Technology, Haifa 32000, Israel

Krzysztof Burdzy
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195-4350

Keywords: Neumann eigenfunctions, reflected Brownian motion, couplings, hot spots problem
Received by editor(s): December 17, 2001
Published electronically: February 11, 2004
Additional Notes: Research partially supported by the fund for the promotion of research at the Technion
The second author gratefully acknowledges the hospitality and financial support of Technion (Israel) and Institut Mittag-Leffler (Sweden). This research was partially supported by NSF Grant DMS-0071486 and ISF Grant 12/98
Article copyright: © Copyright 2004 American Mathematical Society