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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Auchmuty G., Bao D.: Harnack-Type Inequalities for Evolution Equations. Proc. Amer. Math. Soc. 122, 117–129 (1994)
Aronson D.G., Serrin J.: Local Behaviour of Solutions of Quasilinear Parabolic Equations. Arch. Rational Mech. Anal. 25, 81–122 (1967)
Barenblatt G.I.: On some unsteady motions of a liquid or a gas in a porous medium. Prikl. Mat. Mech. 16, 67–78 (1952)
Barenblatt G.I.: On self-similar motions of a compressible fluid in a porous medium. Prikl. Mat. Mech. 16, 679–698 (1952)
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.
Bombieri E., Giusti E.: Harnack’s inequality for elliptic differential equations on minimal surfaces. Inventiones Math. 15, 24–46 (1972)
Bonforte M., Vàzquez J. L.: Positivity, local smoothing and Harnack inequalities for very fast diffusion equations. Adv. Math. 223, 529–578 (2010)
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)
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.
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.
Chen Y.Z., DiBenedetto E.: Hölder estimates of solutions of singular parabolic equations with measurable coefficients. Arch. Rat. Mech. Anal. 118, 257–271 (1992)
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.
E. DiBenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J. 32 (1983), 83–118.
DiBenedetto E.: On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients. Ann. Sc. Norm. Sup. 13, 487–535 (1986)
DiBenedetto E.: Intrinsic Harnack type inequalities for solutions of certain degenerate parabolic equations. Arch. Rational Mech. Anal. 100, 129–147 (1988)
DiBenedetto E.: Harnack Estimates in Certain Function Classes. Atti Sem. Mat. Fis. Univ. Modena 37, 173–182 (1989)
E. DiBenedetto, Degenerate parabolic equations, Springer Verlag, 1993.
E. DiBenedetto, U. Gianazza, and V. Vespri, Harnack estimates for quasilinear degenerate parabolic differential equations, Acta Math. 200 (2008), 181–209.
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)
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.
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.
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.
E. DiBenedetto, U. Gianazza and V. Vespri, Harnack’s inequality for degenerate and singular parabolic equations, Springer Monographs in Mathematics, 2011.
DiBenedetto E., Kwong Y.C.: Intrinsic Harnack Estimates and Extinction Profile for Certain Singular Parabolic Equations, Trans. Amer. Math. Soc. 330, 783–811 (1992)
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.
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.
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.
Fabes E.B., Garofalo N.: Parabolic B.M.O. and Harnack’s inequality. Proc. Amer. Math. Soc. 95, 63–69 (1985)
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.
W. Feller, Über die Lösungen der linearen partiellen Differentialgleichungen zweiter Ordnung vom elliptischen Typus, Math. Ann. 102 (1930), 633–649.
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.
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.
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)
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.
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.
Gianazza U., Vespri V.: A Harnack inequality for a degenerate parabolic equation. J. Evol. Equs. 6, 247–267 (2006)
J. Hadamard, Extension à l’equation de la chaleur d’un théorème de A. Harnack, Rend. Circ. Mat. Palermo 3 (1954), 337–346.
C. G. A. von Harnack, Die Grundlagen der Theorie des logaritmischen Potentiales und der eindeutigen Potentialfunktion in der Ebene, Teubner, Leipzig, Germany, 1887.
John F., Nirenberg L.: On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14, 415–426 (1961)
M. Kassmann, Harnack Inequalities: An Introduction, Bound. Value Probl. 2007, Art. ID 81415, 21 pages.
T. Kuusi, Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008), 673–716.
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.
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.
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.
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.
P. Li and S.-T.Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 153–201.
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.
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)
Moser J.: On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14, 577–591 (1961)
Moser J.: A Harnack Inequality for Parabolic Differential Equations. Comm. Pure Appl. Math. 17, 101–134 (1964)
J. Moser,Correction to: A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 20 (1967), 231–236.
J. Moser, On a pointwise estimate for parabolic differential equations, Comm. Pure Appl. Math. 24 (1971), 727–740.
J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954.
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.
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)
L. Saloff-Coste, Aspects of SobolevType Inequalities, London Mathematical Society Lecture Notes Series, 289, 2002.
J. Serrin, On the Harnack inequality for linear elliptic equations, J. Analyse Math. 4 (1955-56), 292–308.
Serrin J.: Local behavior of solutions of quasilinear equations. Acta Math. 111, 247–302 (1964)
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.
Trudinger N.S.: On Harnack type inequalities and their application to quasilinear elliptic partial differential equations. Comm. Pure Appl. Math. 20, 721–747 (1967)
Trudinger N.S.: Pointwise Estimates and Quasilinear Parabolic Equations. Comm. Pure Appl. Math. 21, 205–226 (1967)
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).
Author information
Authors and Affiliations
Corresponding author
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
About this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00032-015-0240-3