On a conjecture of Laugesen and Morpurgo

TL;DR

This paper proves a conjecture about the Neumann heat kernel in the unit ball using probabilistic methods and provides a new proof of the Hot Spots conjecture for the unit disk.

Contribution

It offers a probabilistic proof of Laugesen and Morpurgo's conjecture and applies this to reprove Rauch's Hot Spots conjecture for the unit disk.

Findings

The diagonal element of the Neumann heat kernel is radially increasing.

The conjecture of Laugesen and Morpurgo is settled.

A new proof of the Hot Spots conjecture for the unit disk is provided.

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 $\mathbb{R}^{n}$ ($n\geq1$) 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.