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Local behaviour of solutions to doubly nonlinear parabolic equations

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Abstract

We give a relatively simple and transparent proof for Harnack’s inequality for certain degenerate doubly nonlinear parabolic equations. We consider the case where the Lebesgue measure is replaced with a doubling Borel measure which supports a Poincaré inequality.

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References

  1. Aronsson D.G., Serrin J. (1967) Local behaviour of solutions of quasilinear parabolic equations. Arch. Rat. Mech. Anal 25, 81–122

    Article  Google Scholar 

  2. Bakry D., Coulhon T., Ledoux M., Saloff-Coste L. (1995) Sobolev inequalities in disguise. Indiana Univ. Math. J. 44:1033–1074

    Article  MathSciNet  Google Scholar 

  3. Bombieri, E.: Theory of minimal surfaces and a counterexample to the bernstein conjecture in high dimension, Mimeographed Notes of Lectures Held at Courant Institute, New York University (1970)

  4. Bombieri E., Giusti E. (1972) Harnack’s inequality for elliptic differential equations on minimal surfaces. Invent. Math. 15, 24–46

    Article  MATH  MathSciNet  Google Scholar 

  5. Chiarenza F., Frasca M. (1985) A note on weighted Sobolev inequality. Proc. Am. Math. Soc. 93, 703–704

    Article  MATH  MathSciNet  Google Scholar 

  6. Chiarenza F., Serapioni R. (1984) A Harnack inequality for degenerate parabolic equations. Comm. Partial Differ. Equ. 9(8):719–749

    Article  MATH  MathSciNet  Google Scholar 

  7. Coulhon T., Holopainen I., Saloff-Coste L. (2001) Harnack inequality and hyperbolicity for subelliptic p-Laplacians with applications to Picard type theorems. Geom. Funct. Anal. 11(6):1139–1191

    Article  MATH  MathSciNet  Google Scholar 

  8. DiBenedetto E. (1993) Degenerate Parabolic Equations. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  9. DiBenedetto E., Urbano J.M., Vespri V. (2004) Current issues on singular and degenerate evolution equations. Handbook of Differential Equations, pp. 169–286. Elsevier

    Google Scholar 

  10. Fabes E., Garofalo N. (1985) Parabolic B.M.O. and Harnack’s inequality. Proc. Am. Math. Soc. 50(1):63–69

    Article  MathSciNet  Google Scholar 

  11. Fabes E., Kenig C., Serapioni R. (1982) The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differ. Equ. 7(1):77–116

    Article  MATH  MathSciNet  Google Scholar 

  12. Gianazza, U., Vespri, V.: A Harnack inequality for solutions of doubly nonlinear parabolic equation. J. Appl. Funct. Anal (to appear)

  13. Giaquinta, M.: Introduction to Regularity Theory for Nonlinear Elliptic Systems. Birkhäuser Verlag (1993)

  14. Grigor’yan, A.: The heat equation on non-compact Riemannian manifolds. Matem. Sbornik 182, 55–87 (1991). Engl. Transl. Math. USSR Sb. 72, 47–77 (1992)

    Google Scholar 

  15. Hajł asz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 688 (2000)

  16. Heinonen, J., Kilpeläinen, T., Martio O.: Nonliear Potential Theory of Degenerate Elliptic Equations Oxford University Press, Oxford (1993)

  17. Kinnunen J., Shanmugalingam N. (2001) Regularity of quasi-minimizers on metric spaces. Manuscripta Math. 105, 401–423

    Article  MATH  MathSciNet  Google Scholar 

  18. Moser, J.: A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17, 101–134 (1964), and correction in Comm. Pure Appl. Math. 20, 231–236 (1967)

  19. Moser J. (1971) On a pointwise estimate for parabolic equations. Comm. Pure Appl. Math. 24, 727–740

    Article  MATH  MathSciNet  Google Scholar 

  20. Saloff-Coste, L.: Aspects of Sobolev-type inequalities, London Mathematical Society Lecture Note Series, vol. 289. Cambridge University Press (2002)

  21. Saloff-Coste L. (1992) A note on Poincaré, Sobolev and Harnack inequalities. Duke Math. J. 65, (IMRN 2):27–38

    MathSciNet  Google Scholar 

  22. Trudinger N.S. (1968) Pointwise estimates and quasilinear parabolic equations. Comm. Pure Appl. Math. 21, 205–226

    Article  MATH  MathSciNet  Google Scholar 

  23. Vespri V. (1992) On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations. Manuscripta Math. 75, 65–80

    Article  MATH  MathSciNet  Google Scholar 

  24. Wu Z., Zhao J., Yin J., Li H. (2001) Nonlinear Diffusion Equations. World Scientific, Singapore

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014, Oulu, Finland

    Juha Kinnunen

  2. Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, 02015, HUT, Finland

    Tuomo Kuusi

Authors
  1. Juha Kinnunen
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  2. Tuomo Kuusi
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Corresponding author

Correspondence to Juha Kinnunen.

Additional information

The research is supported by the Finnish Academy of Science and Letters, the Vilho, Yrjö and Kalle Väisälä Foundation.

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Kinnunen, J., Kuusi, T. Local behaviour of solutions to doubly nonlinear parabolic equations. Math. Ann. 337, 705–728 (2007). https://doi.org/10.1007/s00208-006-0053-3

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  • DOI: https://doi.org/10.1007/s00208-006-0053-3

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