On a conjecture of Laugesen and Morpurgo

Abstract

A well known conjecture of R. Laugesen and C. Morpurgo asserts that the diagonal element of the Neumann heat kernel of the unit ball in ${\mathbb{R}}^n$ ($n\ge 1$) is a radially increasing function. In this paper, we use probabilistic arguments to settle this conjecture, and, as an application, we derive a new proof of the Hot Spots conjecture of J. Rauch in the case of the unit disk.

14 pages, 1 figure