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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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On Neumann eigenfunctions in lip domains
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by Rami Atar and Krzysztof Burdzy PDF
J. Amer. Math. Soc. 17 (2004), 243-265 Request permission

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
  • Email: atar@ee.technion.ac.il
  • Krzysztof Burdzy
  • Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195-4350
  • Email: burdzy@math.washington.edu
  • 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
  • © Copyright 2004 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 17 (2004), 243-265
  • MSC (2000): Primary 35J05; Secondary 60H30
  • DOI: https://doi.org/10.1090/S0894-0347-04-00453-9
  • MathSciNet review: 2051611