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Potential Analysis

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

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

  • M. van den Berg
Article

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.

Keywords

Heat flow Perimeter Euclidean space 

Mathematics Subject Classification (2010)

35K05 

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References

  1. 1.
    van den Berg, M.: Heat content and Brownian motion for some regions with a fractal boundary. Probab. Theory Relat. Fields 100, 439–456 (1994)CrossRefMATHGoogle Scholar
  2. 2.
    van den Berg, M.: Heat content asymptotics for planar regions with cusps. J. Lond. Math. Soc. 57, 677–693 (1998)CrossRefGoogle Scholar
  3. 3.
    van den Berg, M., Davies, E.B.: Heat flow out of regions in ℝm. Math. Zeit. 202, 463–482 (1989)CrossRefMATHGoogle Scholar
  4. 4.
    van den Berg, M., Gilkey, P.: Heat content asymptotics of a Riemannian manifold with boundary. J. Funct. Anal. 120, 48–71 (1994)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Evans, L. C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Chapman & Hall / CRC, Boca Raton (1992)MATHGoogle Scholar
  6. 6.
    Gilkey, P.B.: Asymptotic Formulae in Spectral Geometry. Chapman & Hall / CRC, Boca Raton (2004)MATHGoogle Scholar
  7. 7.
    Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence, Rhode Island (1997)Google Scholar
  8. 8.
    Miranda, M. Jr., Pallara, D., Paronetto, F., Preunkert, M.: On a characterisation of perimeters in ℝN via heat semigroup. Ric. Mat. 44, 615–621 (2005)MathSciNetGoogle Scholar
  9. 9.
    Miranda, M. Jr., Pallara, D., Paronetto, F., Preunkert, M.: Short-time heat flow and functions of bounded variation in ℝN. Ann. Fac. Sci. Toulouse 16, 125–145 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Preunkert, M.: A Semigroup version of the isoperimetric inequality. Semigroup Forum 68, 233–245 (2004)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.School of MathematicsUniversity of BristolBristolUK

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