Heat Flow and Perimeter in \(\boldsymbol{{\mathbb{R}}^m}\)

Abstract

Let Ω be an open set in Euclidean space ℝ ^m with finite perimeter \({\mathcal{P}}(\Omega),\) and with m-dimensional Lebesgue measure |Ω|. It was shown by M. Preunkert that if T(t) is the heat semigroup on L ²(ℝ ^m ) then \(H_{\Omega}(t):=\int_{\Omega}T(t)\textbf{1}_{\Omega}(x)dx=|\Omega|-\pi^{-1/2}{\mathcal{P}}(\Omega)t^{1/2}+o(t^{1/2}), \ t\downarrow 0\). H _Ω(t) represents the amount of heat in Ω if Ω is at initial temperature 1 and if ℝ ^m  ∖ Ω is at initial temperature 0. In this paper we will compare the quantitative behaviour of H _Ω(t) with the usual heat content Q _Ω(t) associated to the Dirichlet heat semigroup on Ω. We analyse the heat content for horn-shaped open sets of the form Ω(α, Σ) = {(x, x′) ∈ ℝ ^m : x′ ∈ (1 + x)^− α Σ, x > 0}, where α > 0, and where Σ is an open set in ℝ ^m − 1 with finite perimeter in ℝ ^m − 1, which is star-shaped with respect to 0. For m ≥ 3 we find that there are four regimes with very different behaviour depending on α, and a further two limiting cases where logarithmic corrections appear.