Potential Analysis

, Volume 39, Issue 4, pp 369–387 | Cite as

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

  • M. van den Berg


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.


Heat flow Perimeter Euclidean space 

Mathematics Subject Classification (2010)



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© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.School of MathematicsUniversity of BristolBristolUK

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