Fiber Brownian motion and the ``hot spots'' problem
This article is devoted to the proof of the following negative answer to the so-called ``hot spots'' conjecture: There exists a bounded Lipschitz domain in the plane such that its second Neumann eigenvalue is simple, and both extrema of the corresponding eigenfunction are attained only at interior points of the domain. The domain taken as counterexample is made of two disjoint triangles whose vertices are connected by very thin tubes. The idea (coming from W. Werner) is that if a triangle is warm and the other is cold, then the centers of these triangles should remain respectively the warmest and coldest points of this domain. To proceed, a reflected Brownian motion is run in this domain, and is approximated by a ``fiber Brownian motion'', corresponding to the domain modified by replacing the thin tubes by an uncountable collection of fibers. Such fiber process switches between 1- and 2-dimensional Brownian motions, and yields an approximation of the spectral theory of the domain considered. A lot of delicate technicity is necessry to achieve the arguments.
scientific article; zbMATH DE number 1820753