Journal of Fixed Point Theory and Applications

, Volume 14, Issue 1, pp 267–298 | Cite as

Neumann heat content asymptotics with singular initial temperature and singular specific heat



We study the asymptotic behavior of the heat content on a compact Riemannian manifold with boundary and with singular specific heat and singular initial temperature distributions imposing Robin boundary conditions. Assuming the existence of a complete asymptotic series we determine the first three terms in that series. In addition to the general setting, the interval is studied in detail as are recursion relations among the coefficients and the relationship between the Dirichlet and Robin settings.

Mathematics Subject Classification

58J32 58J35 35K20 


Neumann Laplacian heat content asymptotics singular initial temperature 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    van den Berg M.: Heat equation on the arithmetic von Koch snowflake. Probab. Theory Related Fields 118, 17–36 (2000)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    van den Berg M., Davies E.B.: Heat flow out of regions in \({\mathbb{R}^m}\) . Math. Z. 202, 463–482 (1989)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    M. van den Berg and F. den Hollander, Asymptotics for the heat content of a planar region with a fractal polygonal boundary. Proc. Lond. Math. Soc. (3) 78 (1999), 627–661.Google Scholar
  4. 4.
    van den Berg M., Desjardins S., Gilkey P., Functorality and heat content asymptotics for operators of Laplace type. Topol. Methods Nonlinear Anal. 2 (1993), 147–162.MATHMathSciNetGoogle Scholar
  5. 5.
    M. van den Berg, and J.-F. Le Gall, Mean curvature and the heat equation. Math. Z. 215 (1994), 437–464.Google Scholar
  6. 6.
    van den Berg M., Gilkey P., Heat content asymptotics of a Riemannian manifold with boundary. J. Funct. Anal. 120 (1994), 48–71.CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    M. van den Berg, and P. Gilkey, Heat content asymptotics with singular data. J. Phys. A 45 (2012), 374027.Google Scholar
  8. 8.
    van den Berg M., Gilkey P., Heat invariants for odd-dimensional hemispheres. Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 187–193.CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    van den Berg M., Gilkey P., The heat equation with inhomogeneous Dirichlet conditions. Comm. Anal. Geom. 7 (1999), 279–294.MATHMathSciNetGoogle Scholar
  10. 10.
    M. van den Berg, P. Gilkey, A. Grigor’yan, and K. Kirsten, Hardy inequality and heat semigroup estimates for Riemannian manifolds with singular data. Comm. Partial Differential Equations 37 (2012), 885–900.Google Scholar
  11. 11.
    van den Berg M., Gilkey P., Kirsten K., Growth of heat trace and heat content asymptotic coefficients. J. Funct. Anal. 261 (2011), 2293–2322.CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    M. van den Berg, P. Gilkey, K. Kirsten and V. A. Kozlov, Heat content asymptotics for Riemannian manifolds with Zaremba boundary conditions. Potential Anal. 26 (2007), 225–254.Google Scholar
  13. 13.
    M. van den Berg, P. Gilkey and R. Seeley, Heat content asymptotics with singular initial temperature distributions. J. Funct. Anal. 254 (2008), 3093–3122.Google Scholar
  14. 14.
    van den Berg M., Srisatkunarajah S., Heat flow and Brownian motion for a region in R2 with a polygonal boundary. Probab. Theory Related Fields 86 (1990), 41–52.CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    M. V. Berry and C. J. Howls, High orders of the Weyl expansion for quantum billiards: Resurgence of periodic orbits and the Stokes phenomenon. Proc. R. Soc. Lond. Ser. A 447 (1994), 527–555.Google Scholar
  16. 16.
    H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids. The Clarendon Press, Oxford University Press, New York, 1988.Google Scholar
  17. 17.
    S. Desjardins and P. Gilkey, Heat content asymptotics for operators of Laplace type with Neumann boundary conditions. Math. Z. 215 (1994), 251–268.CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    P. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem. 2nd ed., Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.Google Scholar
  19. 19.
    P. Gilkey, The heat content asymptotics for variable geometries. J. Phys. A 32 (1999), 2825–2834.Google Scholar
  20. 20.
    P. Gilkey and K. Kirsten, Heat content asymptotics with transmittal and transmission boundary conditions. J. Lond. Math. Soc. (2) 68 (2003), 431–443.Google Scholar
  21. 21.
    P. Gilkey, K. Kirsten and J. H. Park, Heat content asymptotics for oblique boundary conditions. Lett. Math. Phys. 59 (2002), 269–276.CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    P. Gilkey and R. J. Miatello, Growth of heat trace coefficients for locally symmetric spaces. J. Math. Phys. 53 (2012), 103506.Google Scholar
  23. 23.
    P. Gilkey and J. H. Park, Heat content asymptotics of an inhomogeneous time dependent process. Modern Phys. Lett. A 15 (2000), 1165–1179.Google Scholar
  24. 24.
    P. Gilkey, J. H. Park and K. Sekigawa, Universal curvature identities III. Int. J. Geom. Methods Mod. Phys. 10 (2013), 1350025.CrossRefMathSciNetGoogle Scholar
  25. 25.
    I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products. Academic Press, New York, 2007.Google Scholar
  26. 26.
    D. M. McAvity, Surface energy from heat content asymptotics. J. Phys. A 26 (1993), 823–830.CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    P. McDonald and R. Meyers, Dirichlet spectrum and heat content. J. Funct. Anal. 200 (2003), 150–159.Google Scholar
  28. 28.
    C. G. Phillips and K. M. Jansons, The short-time transient of diffusion outside a conducting body. Proc. R. Soc. Lond. Ser. A 428 (1990), 431–449.Google Scholar
  29. 29.
    A. Savo, On the asymptotic series of the heat content. In: Global Differential Geometry: TheMathematical Legacy of Alfred Gray (Bilbao, 2000), Contemp. Math. 288, 2001, 428–432.Google Scholar
  30. 30.
    Savo A.: Uniform estimates and the whole asymptotic series of the heat content on manifolds. Geom. Dedicata 73, 181–214 (1998)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    I. Travěnec and L. Šamaj, High orders of Weyl series for the heat content. Proc. R. Soc. Lond. Ser. A 467 (2011), 2479–2499.Google Scholar
  32. 32.
    H. Weyl, The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, NJ, 1939.Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.School of MathematicsUniversity of BristolBristolUnited Kingdom
  2. 2.Mathematics DepartmentUniversity of OregonEugeneUSA
  3. 3.Department of MathematicsKorea Institute for Advanced StudySeoulKorea

Personalised recommendations