GEM - International Journal on Geomathematics

, Volume 9, Issue 2, pp 293–315 | Cite as

Describing the singular behaviour of parabolic equations on cones in fractional Sobolev spaces

  • Stephan Dahlke
  • Cornelia SchneiderEmail author
Original Paper


In this paper, the Dirichlet problem for parabolic equations in a wedge is considered. In particular, we study the smoothness of the solutions in the fractional Sobolev scale \(H^s\), \(s\in \mathbb {R}\). The regularity in these spaces is related with the approximation order that can be achieved by numerical schemes based on uniform grid refinements. Our results provide a first attempt to generalize the well-known \(H^{3/2}\)-Theorem of Jerison and Kenig (J Funct Anal 130:161–219, 1995) to parabolic PDEs. As a special case the heat equation on radial-symmetric cones is investigated.


Parabolic evolution equations Regularity theory Fractional Sobolev spaces 

Mathematics Subject Classification

Primary: 35B65 35K10 Secondary: 46E35 


  1. Bauer, H. F.: Tables of the roots of the associated Legendre function with respect to the degree. Math. Comput. 46(174), 601–602, S29–S41 (1986)MathSciNetCrossRefGoogle Scholar
  2. Cohn, D.L.: Measure Theory. Birkhäuser Advanced Texts: Basel Textbooks, 2nd edn. Springer, New York (2013)CrossRefGoogle Scholar
  3. Dahlke, S., Schneider, C.: Besov regularity of parabolic and hyperbolic PDEs. In: Reine Mathematik, vol. 3, pp. 47. Philipps-University Marburg (2017)Google Scholar
  4. Dahlke, S., Dahmen, W., DeVore, R.: Nonlinear approximation and adaptive techniques for solving elliptic operator equations. In: Dahmen, W., Kurdila, A., Oswald, P. (eds.) Multicale Wavelet Methods for Partial Differential Equations. Wavelet Analysis and Its Application, vol. 6, pp. 237–283. Academic Press, San Diego (1997)CrossRefGoogle Scholar
  5. Dahlke, S., Hansen, M., Schneider, C., Sickel, W.: Basic Properties of Kondratiev spaces, p. 52 (2018a) (Preprint)Google Scholar
  6. Dahlke, S., Hansen, M., Schneider, C., Sickel, W.: On Besov regularity of solutions to nonlinear elliptic partial differential equations, p. 52 (2018b) (Preprint)Google Scholar
  7. Dauge, M.: Elliptic Boundary Value Problems in Corner Domains. Lecture Notes in Mathematics, vol. 1341. Springer, Berlin (1988)CrossRefGoogle Scholar
  8. Dauge, M.: Regularity and singularities in polyhedral domains: the case of Laplace and Maxwell equations. Talk in Karlsruhe. (2008)
  9. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985)zbMATHGoogle Scholar
  10. Grisvard, P.: Singularities in Boundary Value Problems. Research in Applied Mathematics. Springer, Berlin (1992)zbMATHGoogle Scholar
  11. Hackbusch, W.: Ellliptic Differential Equations: Theory and Numerical Treatment. Springer, Berlin (1992)CrossRefGoogle Scholar
  12. Hansen, M., Scharf, B.: Relations between Kondratiev spaces and refined localization Triebel-Lizorkin spaces, p. 24 (2018) (Preprint)Google Scholar
  13. Jerison, D., Kenig, C.E.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, 161–219 (1995)MathSciNetCrossRefGoogle Scholar
  14. Kim, K.-H.: A \(W^p_n\)-theory of parabolic equations with unbounded leading coefficients on non-smooth domains. J. Math. Anal. Appl. 350, 294–305 (2009)MathSciNetCrossRefGoogle Scholar
  15. Kim, K.-H., Krylov, N.V.: On the Sobolev space theory of parabolic and elliptic equations in \(C^1\) domains. SIAM J. Math. Anal. 36(2), 618–642 (2004)MathSciNetCrossRefGoogle Scholar
  16. Kozlov, V.A., Maz’ya, V.G.: Singularities of solutions of the first boundary value problem for the heat equation in domains with conical points II. Izv. Vyssh. Uchebn. Zaved. Mat. 3, 37–44 (1987)MathSciNetGoogle Scholar
  17. Kozlov, V.A., Maz’ya, V.G., Rossmann, J.: Elliptic Boundary Value Problems in Domains with Point Singularities. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  18. Kozlov, V., Nazarov, A.: The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge. Math. Nachr. 287(10), 1142–1165 (2014)MathSciNetCrossRefGoogle Scholar
  19. Kufner, A., Opic, B.: How to define reasonably weighted Sobolev spaces. Comment. Math. Univ. Carolin. 25(3), 537–554 (1984)MathSciNetzbMATHGoogle Scholar
  20. Kufner, A., Opic, B.: Some remarks on the definition of weighted Sobolev spaces. In: Partial Differential Equations (Proceedings of an International Conference), Nauka, Novosibirsk, pp. 120–126 (1986)Google Scholar
  21. Kweon, J.R.: Edge singular behavior for the heat equation on polyhedral cylinders in \(\mathbb{R}^3\). Potential Anal. 38(2), 589–610 (2013)MathSciNetCrossRefGoogle Scholar
  22. Luong, V.T., Loi, D.V.: The first initial-boundary value problem for parabolic equations in a cone with edges. Vestn. St.-Petersbg. Univ. Ser. 1. Mat. Mekh. Asron. 2 60(3), 394–404 (2015)MathSciNetGoogle Scholar
  23. Maz’ya, V.G., Rossmann, J.: Elliptic Equations in Polyhedral Domains, vol. 162. AMS Mathematical Surveys and Monographs, Providence (2010)zbMATHGoogle Scholar
  24. Peetre, J.: A generalization of Courant’s nodal domain theorem. Math. Scand. 5, 15–20 (1957)MathSciNetCrossRefGoogle Scholar
  25. Simon, J.: Sobolev, Besov and Nikol\(^{\prime }\)skiĭ fractional spaces: imbeddings and comparisons for vector valued spaces on an interval. Ann. Mat. Pura Appl. 157(4), 117–148 (1990)MathSciNetCrossRefGoogle Scholar
  26. Taylor, M.E.: Partial Differential Equations I. Basic Theory. Applied Mathematical Sciences, vol. 115, 2nd edn. Springer, New York (2011)Google Scholar
  27. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, vol. 18. North-Holland Mathematical Library, Amsterdam (1978)zbMATHGoogle Scholar
  28. Triebel, H.: Theory of Function Spaces III. Monographs in Mathematics, vol. 100. Birkhäuser, Basel (2006)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.FB12 Mathematik und InformatikPhilipps-Universität MarburgMarburgGermany
  2. 2.Department Mathematik, AM3Friedrich-Alexander-Universität Erlangen-NürnbergErlangenGermany

Personalised recommendations