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Manuscripta Mathematica

, Volume 142, Issue 1–2, pp 187–214 | Cite as

Hölder regularity for parabolic De Giorgi classes in metric measure spaces

  • Mathias MassonEmail author
  • Juhana Siljander
Article

Abstract

We give a proof for the Hölder continuity of functions in the parabolic De Giorgi classes in metric measure spaces. We assume the measure to be doubling, to support a weak (1, p)-Poincaré inequality and to satisfy the annular decay property.

Mathematics Subject Classification (1991)

Primary 35B65 Secondary 35K65 31E05 

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References

  1. 1.
    Anders B., Jana B.: Nonlinear Potential Theory on Metric Spaces, vol. 17 of EMS Tracts in Mathematics. European Mathematics Society (EMS), Zurich (2011)Google Scholar
  2. 2.
    Anders B., Jana B., Nageswari S.: The Diriclet problem for p-harmonic functions on metric spaces. J. Reine Angew. Math. 556, 173–203 (2003)MathSciNetzbMATHGoogle 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.
    Jeff C.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9(3), 428–517 (1999)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chiarenza F., Serapinoi R.: Degenerate parabolic equations and Harnack inequality. Ann. Mat. Pura Appl. 137(4), 139–162 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Chiarenza F.M., Serapioni R.P.: A Harnack inequality for degenerate parabolic equations. Commun. Partial Diff. Equ. 9(8), 719–749 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chiarenza F., Serapioni R.: A remark on a Harnack inequality for degenerate parabolic equations. Rend. Sem. Mat. Univ. Padova 73, 179–190 (1985)MathSciNetzbMATHGoogle Scholar
  8. 8.
    DiBenedetto E.: On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13(3), 487–535 (1986)MathSciNetzbMATHGoogle Scholar
  9. 9.
    DiBenedetto E.: Harnack estimates in certain function classes. Atti Seminario Matematico e Fisico Univ. Modena 37, 173–182 (1988)MathSciNetGoogle Scholar
  10. 10.
    DiBenedetto E.: Degenerate Parabolic Equations. Universitext. Springer, New York (1993)zbMATHCrossRefGoogle Scholar
  11. 11.
    Franchi B., Hajlaz P., Koskela P.: Definitions of Sobolev classes in metric spaces. Ann. Inst. Fourier (Grenoble) 49, 1903–1924 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Gianazza U., Vespri V.: Parabolic De Giorgi classes of order p and the Harnack inequality. Calc. Var. Partial Differ. Equ. 26(3), 379–399 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Giaquinta M.: Introduction to Regularity Theory for Nonlinear Elliptic Systems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (1993)zbMATHGoogle Scholar
  14. 14.
    Giusti E.: Direct Methods in the Calulus of Variations. World Scientific Publishing, River Edge (2003)CrossRefGoogle Scholar
  15. 15.
    Hajlasz P., Koskela P.: Sobolev meets Poincaré. C. R. Acad. Sci. Paris 320(1), 1211–1215 (1995)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Heinonen J.: Lectures on Analysis on Metric Spaces. Universitext. Springer, New York (2001)zbMATHCrossRefGoogle Scholar
  17. 17.
    Heinonen J., Koskela P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181, 1–61 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Keith S., Keith S., Keith S.: The Poincaré inequality is an open ended condition. Ann. Math.(2) 167(2), 575–599 (2008)zbMATHCrossRefGoogle Scholar
  19. 19.
    Kinnunen J., Shanmugalingam N.: Regularity of quasi-minimizers on metric spaces. Manuscr. Math. 105(3), 401–423 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Koskela P., Macmanus P.: Quasiconformal mappings and Sobolev spaces. Studia Math. 131, 1–17 (1998)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kuusi T., Laleoglu R., Siljander J., Urbano J.M.: Hölder continuity for trudinger’s equation in measure spaces. Calc. Var. Partial Differ. Equ. 45(1–2), 193–229 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Kuusi, T., Siljander, J., Urbano, J.M: Local hölder continuity for doubly nonlinear parabolic equations. Indiana Univ. Math. J. (To appear)Google Scholar
  23. 23.
    Ladyzhenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Translation of Mathematical Monographs, vol. 23. American Mathematical Society, Providence (1968)Google Scholar
  24. 24.
    Shanmugalingam N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16(2), 243–279 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Shanmugalingam N.: Harmonic functions on metric spaces. Illinois J. Math. 45, 1021–1050 (2001)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Urbano J.M.: The Method of Intrinsic Scaling, vol. 1930 of Lecture Notes in Mathematics. A Systematic Approach to Regularity for Degenerate and Singular PDEs Springer, Berlin (2008)Google Scholar
  27. 27.
    Wieser W.: Parabolic Q-minima and minimal solutions to variational flows. Manuscr. Math. 59(1), 63–107 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Wu Z., Zhao J., Yin J., Li H.: Nonlinear Diffusion Equation. World Scientific, Singapore (2001)CrossRefGoogle Scholar
  29. 29.
    Zhou S.: On the local behaviour of parabolic Q-minima. J. Partial Differ. Equ. 6(3), 255–272 (1993)zbMATHGoogle Scholar
  30. 30.
    Zhou S.: Parabolic Q-minima and their application. J. Partial Differ. Equ. 7(4), 289–322 (1994)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Mathematics and Systems AnalysisAalto UniversityAaltoFinland
  2. 2.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland

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