On the heat content functional and its critical domains

Abstract

We study and classify smooth bounded domains in an analytic Riemannian manifold which are critical for the heat content at all times $$t>0$$ t > 0 . We do that by first computing the first variation of the heat content, and then showing that $$\Omega $$ Ω is critical if and only if it has the so-called constant flow property, so that we can use a previous classification result established in [33] and [34]. The outcome is that $$\Omega $$ Ω is critical for the heat content at time t, for all $$t>0$$ t > 0 , if and only if $$\Omega $$ Ω 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 $$T_1(\Omega ),T_2(\Omega ),\dots $$ T 1 ( Ω ) , T 2 ( Ω ) , ⋯ , which generalize the torsional rigidity $$T_1$$ T 1 . We prove that $$\Omega $$ Ω is critical for all $$T_k$$ T k if and only if $$\Omega $$ Ω is critical for the heat content at every time t, 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 respects they generalize the properties of the foliation of $$\mathbf{R}^{n}$$ R n by Euclidean spheres.

On the placement of an obstacle so as to optimize the Dirichlet heat content

“…Our proof relies on (weak, strong, and Friedman's) maximum principles for parabolic equations [26,28,29,49,59], Kac's principle of not feeling the boundary, and Savo's variational formula for Dirichlet heat content [62].…”