Nodal geometry, heat diffusion and Brownian motion

Georgiev, Bogdan; Mukherjee, Mayukh

doi:10.2140/apde.2018.11.133

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Nodal geometry, heat diffusion and Brownian motion

Bogdan Georgiev and Mayukh Mukherjee

Vol. 11 (2018), No. 1, 133–148

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 $Ω_{λ} \subseteq M$ almost fully contains a ball of radius $\sim 1 ∕ \sqrt{λ_{1} (Ω_{λ})}$ , and such a ball can be centred at any point of maximum of the Dirichlet ground state $φ_{λ_{1} (Ω_{λ})}$ . 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 $M$ .

Keywords

Laplace eigenfunctions, nodal domains, Brownian motion

Mathematical Subject Classification 2010

Primary: 35P20, 53B20, 53Z05

Secondary: 35K05

Milestones

Received: 11 April 2016

Revised: 20 June 2017

Accepted: 10 August 2017

Published: 17 September 2017

Authors

Bogdan Georgiev
Max Planck Institute for Mathematics Bonn Germany

Mayukh Mukherjee
Max Planck Institute for Mathematics Bonn Germany

Vol. 11, No. 1, 2018