Acta Mathematica

, 200:181

Harnack estimates for quasi-linear degenerate parabolic differential equations

  • Emmanuele DiBenedetto
  • Ugo Gianazza
  • Vincenzo Vespri

DOI: 10.1007/s11511-008-0026-3

Cite this article as:
DiBenedetto, E., Gianazza, U. & Vespri, V. Acta Math (2008) 200: 181. doi:10.1007/s11511-008-0026-3


We establish the intrinsic Harnack inequality for non-negative solutions of a class of degenerate, quasilinear, parabolic equations, including equations of the p-Laplacian and porous medium type. It is shown that the classical Harnack estimate, while failing for degenerate parabolic equations, it continues to hold in a space-time geometry intrinsic to the degeneracy. The proof uses only measure-theoretical arguments, it reproduces the classical Moser theory, for non-degenerate equations, and it is novel even in that context. Hölder estimates are derived as a consequence of the Harnack inequality. The results solve a long standing problem in the theory of degenerate parabolic equations.

2000 Math. Subject Classification

Primary 35K65, 35B65Secondary 35B45


Degenerate parabolic equationsHarnack estimatesHölder continuity

Copyright information

© Institut Mittag-Leffler 2008

Authors and Affiliations

  • Emmanuele DiBenedetto
    • 1
  • Ugo Gianazza
    • 2
  • Vincenzo Vespri
    • 3
  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA
  2. 2.Dipartimento di Matematica “F. Casorati”Università di PaviaPaviaItaly
  3. 3.Dipartimento di Matematica “U. Dini”Università di FirenzeFirenzeItaly