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Hölder continuity for Trudinger’s equation in measure spaces

  • Tuomo Kuusi
  • Rojbin Laleoglu
  • Juhana Siljander
  • José Miguel Urbano
Article

Abstract

We complete the study of the regularity for Trudinger’s equation by proving that weak solutions are Hölder continuous also in the singular case. The setting is that of a measure space with a doubling non-trivial Borel measure supporting a Poincaré inequality. The proof uses the Harnack inequality and intrinsic scaling.

Mathematics Subject Classification (2000)

35B65 35K67 35D10 

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References

  1. 1.
    Acerbi E., Mingione G.: Gradient estimates for a class of parabolic systems. Duke Math. J. 136(2), 285–320 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bögelein V., Duzaar F., Mingione G.: Degenerate problems with irregular obstacles. J. Reine Angew. Math. 650, 107–160 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Buckley S.M.: Is the maximal function of a Lipschitz function continuous?. Ann. Acad. Sci. Fenn. Math. 24(2), 519–528 (1999)MathSciNetGoogle Scholar
  4. 4.
    DiBenedetto E.: Degenerate Parabolic Equations. Universitext. Springer-Verlag, New York (1993)CrossRefGoogle Scholar
  5. 5.
    DiBenedetto E., Friedman A.: Hölder estimates for nonlinear degenerate parabolic systems. J. Reine Angew. Math. 357, 1–22 (1985)MathSciNetzbMATHGoogle Scholar
  6. 6.
    DiBenedetto E., Urbano J.M., Vespri V.: Current issues on singular and degenerate evolution equations. In: Dafermos, C.M. (eds) Evolutionary Equations. Vol. I Handbook of Differential Equations, pp. 169–286. North-Holland, Amsterdam (2004)Google Scholar
  7. 7.
    DiBenedetto E., Gianazza U., Vespri V.: Harnack estimates for quasi-linear degenerate parabolic differential equations. Acta Math. 200(2), 181–209 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    DiBenedetto E., Gianazza U., Vespri V.: Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9(2), 385–422 (2010)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Fornaro S., Gianazza U.: Local properties of non-negative solutions to some doubly non-linear degenerate parabolic equations. Discret. Contin. Dyn. Syst. 26(2), 481–492 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Fornaro S., Sosio M.: Intrinsic Harnack estimates for some doubly nonlinear degenerate parabolic equations. Adv. Differ. Equ. 13(1–2), 139–168 (2008)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Grigor’yan, A.A.: The heat equation on noncompact Riemannian manifolds. Mat. Sb. 182(1), 55–87 (1991). Translation in Math. USSR-Sb. 72(1), 47–77 (1992)Google Scholar
  12. 12.
    Hajlasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145(688), x+101 (2000)Google Scholar
  13. 13.
    Heinonen J., Kilpeläinen T., Martio O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1993)Google Scholar
  14. 14.
    Henriques E., Urbano J.M.: On the doubly singular equation γ (u)t = Δp u. Commun. Partial Differ. Equ. 30(4-6), 919–955 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Ivanov A.V.: Hölder estimates for equations of fast diffusion type. Algebra i Analiz 6(4), 101–142 (1994)Google Scholar
  16. 16.
    Keith S., Zhong X.: The Poincaré inequality is an open ended condition. Ann. Math. 167(2), 575–599 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Kinnunen J., Kuusi T.: Local behaviour of solutions to doubly nonlinear parabolic equations. Math. Ann. 337(3), 705–728 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Kinnunen J., Shanmugalingam N.: Regularity of quasi-minimizers on metric spaces. Manuscripta Math. 105(3), 401–423 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Kuusi T.: Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7(4), 673–716 (2008)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Kuusi, T., Siljander, J., Urbano, J.M.: Hölder continuity to a doubly nonlinear parabolic equation. Indiana Univ. Math. J. (to appear)Google Scholar
  21. 21.
    Porzio M.M., Vespri V.: Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations. J. Differ. Equ. 103(1), 146–178 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Saloff-Coste L.: A note on Poincaré, Sobolev, and Harnack inequalities. Int. Math. Res. Notices 2, 27–38 (1992)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Saloff-Coste L.: Aspects of Sobolev-type Inequalities, Volume 289 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2002)Google Scholar
  24. 24.
    Trudinger N.S.: Pointwise estimates and quasilinear parabolic equations. Commun. Pure Appl. Math. 21, 205–226 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Urbano J.M.: The Method of Intrinsic Scaling. A Systematic Approach to Regularity for Degenerate and Singular PDEs. Volume 1930 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (2008)Google Scholar
  26. 26.
    Vespri V.: On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations. Manuscripta Math. 75(1), 65–80 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Zhou S.: Parabolic Q-minima and their application. J. Partial Differ. Equ. 7(4), 289–322 (1994)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Tuomo Kuusi
    • 1
  • Rojbin Laleoglu
    • 2
  • Juhana Siljander
    • 1
  • José Miguel Urbano
    • 2
  1. 1.Institute of MathematicsAalto UniversityAaltoFinland
  2. 2.CMUC, Department of MathematicsUniversity of CoimbraCoimbraPortugal

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