Skip to main content
SpringerLink
Log in

Interior Harnack Estimates: The State-of-the-Art for Quasilinear Singular Parabolic Equations

  • Published:
Milan Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper we give some historical information about elliptic and parabolic Harnack inequalities. Then we state the main results known for Harnack inequalities of solutions to quasilinear degenerate parabolic equations. Lastly we focus our attention on Harnack inequalities of solutions to quasilinear singular parabolic equations where the theory did important steps forward in the last few years but still there are some points to be fully understood.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Auchmuty G., Bao D.: Harnack-Type Inequalities for Evolution Equations. Proc. Amer. Math. Soc. 122, 117–129 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aronson D.G., Serrin J.: Local Behaviour of Solutions of Quasilinear Parabolic Equations. Arch. Rational Mech. Anal. 25, 81–122 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  3. Barenblatt G.I.: On some unsteady motions of a liquid or a gas in a porous medium. Prikl. Mat. Mech. 16, 67–78 (1952)

    MATH  MathSciNet  Google Scholar 

  4. Barenblatt G.I.: On self-similar motions of a compressible fluid in a porous medium. Prikl. Mat. Mech. 16, 679–698 (1952)

    MATH  MathSciNet  Google Scholar 

  5. E. Bombieri, Theory of minimal surfaces and a counterexample to the Bernstein conjecture in high dimension, Mimeographed Notes of Lectures held at Courant Institut, New York University, 1970.

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Bonforte M., Vàzquez J. L.: Positivity, local smoothing and Harnack inequalities for very fast diffusion equations. Adv. Math. 223, 529–578 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bonforte M., Iagar R.G., V´azquez J.L.: Local smoothing effects, positivity and Harnack inequalities for the fast p-Laplacian equation. Adv. Math. 224, 2151–2215 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. L. A. Caffarelli and A. Friedman, Continuity of the density of a gas flow in a porous medium, Trans. Amer. Math. Soc. 252 (1979), 99–113.

  10. M. Calahorrano Recalde and V. Vespri, Harnack estimates at large: sharp pointwise estimates for nonnegative solutions to a class of singular parabolic equations, Nonlinear Anal. 121 (2015), 153–163.

  11. Chen Y.Z., DiBenedetto E.: Hölder estimates of solutions of singular parabolic equations with measurable coefficients. Arch. Rat. Mech. Anal. 118, 257–271 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  12. Y. Z. Chen and E. DiBenedetto, On the Harnack inequality for nonnegative solutions of singular parabolic equations, Degenerate diffusions (Minneapolis, 1991), 61–69, IMA Vol. Math. Appl. 47 Springer, New York, 1993.

  13. E. DiBenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J. 32 (1983), 83–118.

  14. DiBenedetto E.: On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients. Ann. Sc. Norm. Sup. 13, 487–535 (1986)

    MATH  MathSciNet  Google Scholar 

  15. DiBenedetto E.: Intrinsic Harnack type inequalities for solutions of certain degenerate parabolic equations. Arch. Rational Mech. Anal. 100, 129–147 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  16. DiBenedetto E.: Harnack Estimates in Certain Function Classes. Atti Sem. Mat. Fis. Univ. Modena 37, 173–182 (1989)

    MATH  MathSciNet  Google Scholar 

  17. E. DiBenedetto, Degenerate parabolic equations, Springer Verlag, 1993.

  18. E. DiBenedetto, U. Gianazza, and V. Vespri, Harnack estimates for quasilinear degenerate parabolic differential equations, Acta Math. 200 (2008), 181–209.

  19. DiBenedetto E., Gianazza U., Vespri V.: Sub-Potential Lower Bounds for Nonnegative local solutions to certain quasi linear degenerate parabolic equations. Duke Math. J. 143, 1–15 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. E. DiBenedetto, U. Gianazza and V. Vespri, Alternative Forms of the Harnack Inequality for Non-Negative Solutions to Certain Degenerate and Singular Parabolic Equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. 20 (2009), 369–377.

  21. E. DiBenedetto, U. Gianazza and V. Vespri, Forward, backward and elliptic Harnack inequality for nonnegative solution to certain singular parabolic partial differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. 11 (2010), 385–422.

  22. E. DiBenedetto, U. Gianazza and V. Vespri, Harnack type estimates and Hölder continuity for nonnegative solutions to certain subcritically singular parabolic partial differential equations, Manuscripta Math. 131 (2010), 231–245.

  23. E. DiBenedetto, U. Gianazza and V. Vespri, Harnack’s inequality for degenerate and singular parabolic equations, Springer Monographs in Mathematics, 2011.

  24. DiBenedetto E., Kwong Y.C.: Intrinsic Harnack Estimates and Extinction Profile for Certain Singular Parabolic Equations, Trans. Amer. Math. Soc. 330, 783–811 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  25. E. Di Benedetto, J. M. Urbano and V. Vespri, Current Issues on Singular and Degenerate Evolution Equations, Evolutionary equations. Vol.1, Handb. Differ. Equ., North- Holland, Amsterdam, (2004), 169–286.

  26. E. De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3 (1957), 25–43.

  27. F. G. Düzgün, P. Marcellini and V. Vespri, An alternative approach to the Hölder continuity of solutions to some elliptic equations, NonLinear Anal. 94 (2014), 133–141.

  28. Fabes E.B., Garofalo N.: Parabolic B.M.O. and Harnack’s inequality. Proc. Amer. Math. Soc. 95, 63–69 (1985)

    MATH  MathSciNet  Google Scholar 

  29. E.B. Fabes and D.W. Stroock, A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash, Arch. Rational Mech. Anal. 96 (1986), 327–338.

  30. W. Feller, Über die Lösungen der linearen partiellen Differentialgleichungen zweiter Ordnung vom elliptischen Typus, Math. Ann. 102 (1930), 633–649.

  31. A. Z. Fino, F. G. Düzgün and V.Vespri, Conservation of the mass for solutions to a class of singular parabolic equations, Kodai Math. J. 37 (2014) no. 3, 519–531.

  32. S. Fornaro and V. Vespri, Harnack Estimates for non-negative weak solutions of a class of singular parabolic equations, Manuscripta Math. 141 (2013), 85–103.

  33. Gianazza U., Polidoro S.: Lower bounds for solutions of degenerate parabolic equations, Subelliptic PDEs and applications to geometry and finance. Lect. Notes Semin. Interdiscip. Mat. 6, 157–162 (2007)

    MathSciNet  Google Scholar 

  34. U. Gianazza, M. Surnachev and V. Vespri, On a new proof of Hölder continuity of solutions of p-Laplace type parabolic equations, Adv. Calc. Var. 3 (2010), no. 3, 263–278.

  35. U. Gianazza and V. Vespri, Parabolic De Giorgi classes of order p and the Harnack inequality, Calc. Var. Partial Differential Equations 26 (2006) 379–399.

  36. Gianazza U., Vespri V.: A Harnack inequality for a degenerate parabolic equation. J. Evol. Equs. 6, 247–267 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  37. J. Hadamard, Extension à l’equation de la chaleur d’un théorème de A. Harnack, Rend. Circ. Mat. Palermo 3 (1954), 337–346.

  38. C. G. A. von Harnack, Die Grundlagen der Theorie des logaritmischen Potentiales und der eindeutigen Potentialfunktion in der Ebene, Teubner, Leipzig, Germany, 1887.

  39. John F., Nirenberg L.: On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14, 415–426 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  40. M. Kassmann, Harnack Inequalities: An Introduction, Bound. Value Probl. 2007, Art. ID 81415, 21 pages.

  41. T. Kuusi, Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008), 673–716.

  42. O. A. Ladyzenskaya, N. A. Solonnikov and N.N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968.

  43. O. A. Ladyzenskaya and N.N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc., Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968.

  44. E. Lanconelli, Bruno Pini and the parabolic Harnack inequality: the dawning of Parabolic Potential Theory, Mathematicians in Bologna 1861-1960, Basel, Birkhäuser, 2012, 317–332.

  45. L. von Lichtenstein, Beiträge zur Theorie der linearen partiellen Differentialgleichungen zweiter Ordnung vom elliptischen Typus. Unendliche Folgen positiver Lösungen, Palermo Rend. 33 (1912), 201–211.

  46. P. Li and S.-T.Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 153–201.

  47. J. L. Lions, Quelques problémes de la théorie des équations non linéaires dé’volution, Problems in non-linear analysis, C.I.M.E., IV Ciclo, Varenna (1970), 189–342. Edizioni Cremonese, Rome, 1971.

  48. Moser J.: A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations. Comm. Pure Appl. Math. 13, 457–468 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  49. Moser J.: On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14, 577–591 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  50. Moser J.: A Harnack Inequality for Parabolic Differential Equations. Comm. Pure Appl. Math. 17, 101–134 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  51. J. Moser,Correction to: A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 20 (1967), 231–236.

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

  53. J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954.

  54. B. Pini, Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico, Rend. Sem. Mat. Univ. Padova 23 (1954), 422–434.

  55. Ragnedda F., Vernier Piro S., Vespri V.: Pointwise estimates for the fundamental solutions of a class of singular parabolic problems. J. Anal. Math. 121, 235–253 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  56. L. Saloff-Coste, Aspects of SobolevType Inequalities, London Mathematical Society Lecture Notes Series, 289, 2002.

  57. J. Serrin, On the Harnack inequality for linear elliptic equations, J. Analyse Math. 4 (1955-56), 292–308.

  58. Serrin J.: Local behavior of solutions of quasilinear equations. Acta Math. 111, 247–302 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  59. I. I. Skrypnik and A. F. Tedeev, Decay of the mass of the solution to the Cauchy problem of the degenerate parabolic equation with nonlinear potential, in preparation.

  60. Trudinger N.S.: On Harnack type inequalities and their application to quasilinear elliptic partial differential equations. Comm. Pure Appl. Math. 20, 721–747 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  61. Trudinger N.S.: Pointwise Estimates and Quasilinear Parabolic Equations. Comm. Pure Appl. Math. 21, 205–226 (1967)

    Article  MathSciNet  Google Scholar 

  62. J. M. Urbano, The method of intrinsic scaling. A systematic approach to regularity for degenerate and singular PDEs. Lecture Notes in Mathematics, 1930. Springer-Verlag, Berlin (2008).

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria dell’Informazione, Università degli studi di Firenze, Via di Santa Marta, 3, I-50139, Firenze, Italy

    Fatma Gamze Düzgüun

  2. Department of Mathematics, Hacettepe University, 06800, Beytepe, Ankara, Turkey

    Fatma Gamze Düzgüun

  3. Dipartimento di Matematica“Felice Casorat”, Università degli studi di Pavia, via Ferrata, 1, I-27100, Pavia, Italy

    Simona Fornaro

  4. Dipartimento di Matematica ed Informatica “Ulisse Dini”, Università degli studi di Firenze, Viale Morgagni, 67/a, I-50134, Firenze, Italy

    Vincenzo Vespri

Authors
  1. Fatma Gamze Düzgüun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Simona Fornaro
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vincenzo Vespri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vincenzo Vespri.

Additional information

The second and third authors are members of GNAMPA (INdAM).

Lecture given in the Seminario Matematico e Fisico di Milano on March 28, 2014

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Düzgüun, F.G., Fornaro, S. & Vespri, V. Interior Harnack Estimates: The State-of-the-Art for Quasilinear Singular Parabolic Equations. Milan J. Math. 83, 371–395 (2015). https://doi.org/10.1007/s00032-015-0240-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00032-015-0240-3

Mathematics Subject Classification

Keywords

Search

Navigation