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Harnack type estimates and Hölder continuity for non-negative solutions to certain sub-critically singular parabolic partial differential equations

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Abstract

A two-parameter family of Harnack type inequalities for non-negative solutions of a class of singular, quasilinear, homogeneous parabolic equations is established, and it is shown that such estimates imply the Hölder continuity of solutions. These classes of singular equations include p-Laplacean type equations in the sub-critical range \({1 < p \le\frac{2N}{N+1}}\) and equations of the porous medium type in the sub-critical range \({0 < m \le\frac{(N-2)_+}{N}}\).

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Authors and Affiliations

  1. Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN, 37240, USA

    Emmanuele DiBenedetto

  2. Dipartimento di Matematica “F. Casorati”, Università di Pavia, via Ferrata 1, 27100, Pavia, Italy

    Ugo Gianazza

  3. Dipartimento di Matematica “U. Dini”, Università di Firenze, viale Morgagni 67/A, 50134, Firenze, Italy

    Vincenzo Vespri

Authors
  1. Emmanuele DiBenedetto
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  2. Ugo Gianazza
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  3. Vincenzo Vespri
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Correspondence to Vincenzo Vespri.

Additional information

This work has been partially supported by I.M.A.T.I.-C.N.R.-Pavia. DiBenedetto’s work partially supported by NSF grant DMS-0652385.

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DiBenedetto, E., Gianazza, U. & Vespri, V. Harnack type estimates and Hölder continuity for non-negative solutions to certain sub-critically singular parabolic partial differential equations. manuscripta math. 131, 231–245 (2010). https://doi.org/10.1007/s00229-009-0317-9

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  • DOI: https://doi.org/10.1007/s00229-009-0317-9

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