Harnack estimates for quasi-linear degenerate parabolic differential equations

Emmanuele DiBenedetto Department of Mathematics, Vanderbilt University Ugo Gianazza Dipartimento di Matematica “F. Casorati”, Università di Pavia Vincenzo Vespri Dipartimento di Matematica “U. Dini”, Università di Firenze

TBD mathscidoc:1701.331993

Acta Mathematica, 200, (2), 181-209, 2006.2
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.
Degenerate parabolic equations; Harnack estimates; Hölder continuity
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@inproceedings{emmanuele2006harnack,
  title={Harnack estimates for quasi-linear degenerate parabolic differential equations},
  author={Emmanuele DiBenedetto, Ugo Gianazza, and Vincenzo Vespri},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203351834808702},
  booktitle={Acta Mathematica},
  volume={200},
  number={2},
  pages={181-209},
  year={2006},
}
Emmanuele DiBenedetto, Ugo Gianazza, and Vincenzo Vespri. Harnack estimates for quasi-linear degenerate parabolic differential equations. 2006. Vol. 200. In Acta Mathematica. pp.181-209. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203351834808702.
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