Neumann eigenfunctions and Brownian couplings. (English) Zbl 1118.35017

Aikawa, Hiroaki (ed.) et al., Potential theory in Matsue. Selected papers of the international workshop on potential theory, Matsue, Japan, August 23–28, 2004. Tokyo: Mathematical Society of Japan (ISBN 4-931469-33-7/hbk). Advanced Studies in Pure Mathematics 44, 11-23 (2006).
The author studies the hot spots conjecture, namely, that the first non-trivial Neumann eigenfunction of the Laplacian on a bounded Euclidean domain reaches its maximum at the boundary. Explicitly known sufficient conditions for this conjecture to hold are presented. Some probabilistic techniques, in particular the coupling methods, are introduced for the study of the conjecture.
For the entire collection see [Zbl 1102.31001].


35P05 General topics in linear spectral theory for PDEs
35R60 PDEs with randomness, stochastic partial differential equations
60J65 Brownian motion
60H30 Applications of stochastic analysis (to PDEs, etc.)