On the heat content functional and its critical domains

TL;DR

This paper characterizes domains in Riemannian manifolds that are critical for heat content at all times, linking them to isoparametric foliations and extending the understanding of variational properties in PDEs.

Contribution

It provides a complete classification of critical domains for heat content and exit-time moments, connecting these to isoparametric foliations and their variational properties.

Findings

Domains critical for heat content have isoparametric foliations.

Critical domains for exit-time moments coincide with those for heat content.

The work extends the understanding of isoparametric foliations in PDE theory.

Abstract

We study and classify smooth bounded domains in an analytic Riemannian manifold which are critical for the heat content at all times t>0. We do that by first computing the first variation of the heat content, and then showing that a domain is critical if and only if it has the so-called constant flow property, so that we can use a previous classification result established by the author. The outcome is that a domain is critical for the heat content at all times if and only if it admits an isoparametric foliation, that is, a foliation whose leaves are all parallel to the boundary and have constant mean curvature. Then, we consider the sequence of functionals given by the exit-time moments, which generalize the torsional rigidity. We prove that a domain is critical for all exit time moments if and only it is critical for the heat content at all times, and then we get a classification as well. The main purpose of the paper is to understand the variational properties of general isoparametric foliations and their role in PDE's theory; in some respect isoparametric foliations generalize the properties of the foliation of Euclidean space by round spheres.