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)^− αΣ, 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.