On Neumann eigenfunctions in lip domains

Atar, Rami; Burdzy, Krzysztof

doi:10.1090/S0894-0347-04-00453-9

On Neumann eigenfunctions in lip domains
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by Rami Atar and Krzysztof Burdzy

J. Amer. Math. Soc. 17 (2004), 243-265

DOI: https://doi.org/10.1090/S0894-0347-04-00453-9

Published electronically: February 11, 2004

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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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