On the ``hot spots'' conjecture of J. Rauch
type: Item
instance of: scholarly article
title: On the ``hot spots'' conjecture of J. Rauch
zbMATH Open document ID: 0938.35045
author: Rodrigo Bañuelos
author: K. Burdzy
published in: Journal of Functional Analysis
publication date: 2000-06-22T00:00:00Z
full work available at URL: http://hdl.handle.net/1773/2195
review text: The conjecture in the title states roughly, that for most initial data \(u_0\) which are subject to the heat equation under no flux on the boundary of a bounded open connected subset of \(\mathbb{R}^n\), the hot spot moves eventually to the boundary. If one uses separations of variables, this movement of the hot spot can be explained if the (or a) first nonconstant eigenfunction of the Laplacian under Neumann boundary conditions attains its maximum (or maximum modulus) on the boundary (only). The brackets indicate slightly different variants. The authors considerably generalize earlier results of the reviewer [Lecture Notes in Mathematics, 1150. Berlin-Springer (1985; Zbl 0593.35002)] and give various sufficient conditions which imply the monotonicity of the eigenfunctions in certain directions and thus the conjecture. If a domain is long and thin (and this is stated in a precise way) for instance, the eigenfunction is monotone in the long direction. The proofs are partly probabilistic and the paper offers a lot of results, insight and new conjectures about the shape of these eigenfunctions.
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35B50
zbMATH DE Number: 1310878
zbMATH Keywords: eigenfunction
zbMATH Keywords: shape
zbMATH Keywords: hot spot
MaRDI profile type: Publication
OpenAlex ID: W2005045710
cites work: Q3876353
cites work: On domain monotonicity of the Neumann heat kernel
cites work: Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains
cites work: Configurational transition in a Fleming - Viot-type model and probabilistic interpretation of Laplacian eigenfunctions
cites work: Efficient Markovian couplings: Examples and counterexamples
cites work: A counterexample to the ``hot spots'' conjecture
cites work: Q3344779
cites work: Eigenvalue comparison theorems and its geometric applications
cites work: Q4041031
cites work: Applications of analysis on nilpotent groups to partial differential equations
cites work: Q4039796
cites work: Rearrangements and convexity of level sets in PDE
cites work: On the second eigenfunctions of the Laplacian in \({\mathbb{R}}^ 2\)
cites work: Q4720114
cites work: MULTIPLE EIGENVALUES OF THE LAPLACE OPERATOR
cites work: Q4185363
cites work: Spectral gaps and rates to equilibrium for diffusions in convex domains
cites work: Maximum principles and their applications
cites work: Brownian motion in a wedge with oblique reflection
DOI: 10.1006/JFAN.1999.3397
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description: scientific article; zbMATH DE number 1310878