A Counterexample to the "Hot Spots" Conjecture

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 01 Jan 1999Embargo end date: 01 Mar 1998 United States Publisher:JSTORJournal:The Annals of Mathematics, volume 149, page 309 (issn: 0003-486X,

Copyright policy )Funded by:NSF | Brownian Motion and Relat...

Authors: Burdzy, Krzysztof; Werner, Wendelin;

doi: 10.2307/121027 , 10.48550/arxiv.math/9803030

arXiv: math/9803030

handle: 1773/2198

A Counterexample to the "Hot Spots" Conjecture

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Abstract

We construct a counterexample to the ``hot spots'' conjecture; there exists a bounded connected planar domain (with two holes) such that the second eigenvalue of the Laplacian in that domain with Neumann boundary conditions is simple and such that the corresponding eigenfunction attains its strict maximum at an interior point of that domain.

9 pages, published version

Country

United States

Related Organizations

Washington University in St. Louis
United States
University of Washington, University libraries
United States

Keywords

Neumann boundary conditions, Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation, Probability (math.PR), eigenvalues, Estimates of eigenvalues in context of PDEs, Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs, Mathematics - Analysis of PDEs, 35P99 (Primary) 60J65 (Secondary), FOS: Mathematics, Hot spots, eigenfunction, Laplacian, second eigenvalue of the Laplacian, Mathematics - Probability, Analysis of PDEs (math.AP)

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