Harnack and Pointwise Estimates for Degenerate or Singular Parabolic Equations

  • Fatma Gamze Düzgün
  • Sunra MosconiEmail author
  • Vincenzo Vespri
Part of the Springer INdAM Series book series (SINDAMS, volume 33)


In this paper we give both a historical and technical overview of the theory of Harnack inequalities for nonlinear parabolic equations in divergence form. We start reviewing the elliptic case with some of its variants and geometrical consequences. The linear parabolic Harnack inequality of Moser is discussed extensively, together with its link to two-sided kernel estimates and to the Li-Yau differential Harnack inequality. Then we overview the more recent developments of the theory for nonlinear degenerate/singular equations, highlighting the differences with the quadratic case and introducing the so-called intrinsic Harnack inequalities. Finally, we provide complete proofs of the Harnack inequalities in some paramount case to introduce the reader to the expansion of positivity method.


Degenerate and singular parabolic equations Pointwise estimates Harnack estimates Weak solutions Intrinsic geometry 

2010 Mathematics Subject Classification

35K67 35K92 35K20 



We would like to thank an anonymous referee for helping us improve the quality of a first version of the paper. S. Mosconi and V. Vespri are members of GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica). F. G. Düzgün is partially funded by Hacettepe University BAP through project FBI-2017-16260; S. Mosconi is partially funded by the grant PdR 2016–2018 - linea di intervento 2: “Metodi Variazionali ed Equazioni Differenziali” of the University of Catania.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Fatma Gamze Düzgün
    • 1
  • Sunra Mosconi
    • 2
    Email author
  • Vincenzo Vespri
    • 3
  1. 1.Department of MathematicsHacettepe UniversityAnkaraTurkey
  2. 2.Dipartimento di Matematica e InformaticaUniversità degli Studi di CataniaCataniaItaly
  3. 3.Dipartimento di Matematica e Informatica “U. Dini”Università di FirenzeFirenzeItaly

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