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2018 Nodal geometry, heat diffusion and Brownian motion
Bogdan Georgiev, Mayukh Mukherjee
Anal. PDE 11(1): 133-148 (2018). DOI: 10.2140/apde.2018.11.133

Abstract

We use tools from n-dimensional Brownian motion in conjunction with the Feynman–Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold M. On one hand we extend a theorem of Lieb (1983) and prove that any Laplace nodal domain ΩλM almost fully contains a ball of radius 1λ1(Ωλ), and such a ball can be centred at any point of maximum of the Dirichlet ground state φλ1(Ωλ). This also gives a slight refinement of a result by Mangoubi (2008) concerning the inradius of nodal domains. On the other hand, we also prove that no nodal domain can be contained in a reasonably narrow tubular neighbourhood of unions of finitely many submanifolds inside M.

Citation

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Bogdan Georgiev. Mayukh Mukherjee. "Nodal geometry, heat diffusion and Brownian motion." Anal. PDE 11 (1) 133 - 148, 2018. https://doi.org/10.2140/apde.2018.11.133

Information

Received: 11 April 2016; Revised: 20 June 2017; Accepted: 10 August 2017; Published: 2018
First available in Project Euclid: 20 December 2017

zbMATH: 1378.35208
MathSciNet: MR3707293
Digital Object Identifier: 10.2140/apde.2018.11.133

Subjects:
Primary: 35P20, 53B20, 53Z05
Secondary: 35K05

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.11 • No. 1 • 2018
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