Abstract
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.
Keywords
- Degenerate and singular parabolic equations
- Pointwise estimates
- Harnack estimates
- Weak solutions
- Intrinsic geometry
2010 Mathematics Subject Classification
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Actually, to a parabolic version of the Harnack inequality, which readily implies the elliptic one. For further details see the discussion on the parabolic Harnack inequality below and for a nice historical overview on the subject see [80, Section 5.5].
- 2.
Both \(a(s)=\inf I_{s}\) and \(b(s)=\sup I_{s}\) are continuous, hence ∪s ∈ [0,η−1] I s = [infs ∈ [0,η−1] a(s), sups ∈ [0,η−1] b(s)]. Then observe that a(0) = θ(μ)∕2 while b(η − 1) ≥ η θ(μ).
References
D.G. Aronson, Bounds for the fundamental solution of a parabolic equation. Bull. Am. Math. Soc. 73, 890–896 (1967)
D.G. Aronson, J. Serrin, Local behavior of solutions of quasilinear parabolic equations. Arch. Ration. Mech. Anal. 25, 81–122 (1967)
G.I. Barenblatt, On some unsteady motions of a liquid or a gas in a porous medium. Prikl. Mat. Mech. 16, 67–78 (1952)
G.I. Barenblatt, A.S. Monin, Flying sources and the microstructure of the ocean: a mathematical theory. Uspekhi Mat. Nauk. 37, 125–126 (1982)
G.I. Barenblatt, V.M. Entov, V.M. Rizhnik, Motion of Fluids and Gases in Natural Strata (Nedra, Moscow, 1984)
M. Barlow, M. Murugan, Stability of the elliptic Harnack inequality. Ann. Math. 187, 777–823 (2018)
V. Bögelein, F. Duzaar, G. Mingione, The regularity of general parabolic systems with degenerate diffusion. Mem. Am. Math. Soc. 221(1041), x+143 pp. (2013)
V. Bögelein, F. Ragnedda, S. Vernier Piro, V. Vespri, Moser-Nash kernel estimates for degenerate parabolic equations. J. Funct. Anal. 272, 2956–2986 (2017)
E. Bombieri, E. Giusti, Harnack’s inequality for elliptic differential equations on minimal surfaces. Invent. Math. 15, 24–46 (1972)
E. Bombieri, E. De Giorgi, M. Miranda, Una maggiorazione a priori relativa alle ipersuperfici minimali non parametriche. Arch. Rat. Mech. Anal. 32, 255–267 (1965)
M. Bonforte, J.L. Vazquez, Positivity, local smoothing and Harnack inequalities for very fast diffusion equations. Adv. Math. 223, 529–578 (2010)
M. Bonforte, R.G. Iagar, J.L. Vazquez, Local smoothing effects, positivity, and Harnack inequalities for the fast p-Laplacian equation. Adv. Math. 224, 2151–2215 (2010)
M.V. Calahorrano Recalde, V. Vespri, Harnack estimates at large: sharp pointwise estimates for nonnegative solutions to a class of singular parabolic equations. Nonlinear Anal. 121, 153–163 (2015)
M.V. Calahorrano Recalde, V. Vespri, Backward pointwise estimates for nonnegative solutions to a class of singular parabolic equations. Nonlinear Anal. 144, 194–203 (2016)
Y.Z. Chen, E. DiBenedetto, On the local behaviour of solutions of singular parabolic equations. Arch. Ration. Mech. Anal. 103, 319–346 (1988)
S.Y. Cheng, S.T. Yau, Differential equations on Riemannian manifolds and their geometric applications. Commun. Pure Appl. Math. 28, 333–354 (1975)
T.H. Colding, W.P. Minicozzi II, Harmonic functions on manifolds. Ann. Math. 146, 725–747 (1997)
E. De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3, 25–43 (1957)
E. DiBenedetto, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients. Ann. Sc. Norm. Sup. Pisa Cl. Sc. Serie IV 13, 487–535 (1986)
E. DiBenedetto, Intrinsic Harnack type inequalities for solutions of certain degenerate parabolic equations. Arch. Ration. Mech. Anal. 100, 129–147 (1988)
E. DiBenedetto, Degenerate Parabolic Equations. Universitext (Springer, New York, 1993)
E. DiBenedetto, A. Friedman, Hölder estimates for non-linear degenerate parabolic systems. J. Reine Angew. Math. 357, 1–22 (1985)
E. DiBenedetto, U. Gianazza, Some properties of De Giorgi classes. Rend. Istit. Mat. Univ. Trieste 48, 245–260 (2016)
E. DiBenedetto, Y.C. Kwong, Intrinsic Harnack estimates and extinction profile for certain singular parabolic equations. Trans. Am. Math. Soc. 330, 783–811 (1992)
E. DiBenedetto, N.S. Trudinger, Harnack inequalities for quasi-minima of Variational integrals. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 295–308 (1984)
E. DiBenedetto, U. Gianazza, V. Vespri, Harnack Estimates for quasi-linear degenerate parabolic differential equation. Acta Math. 200, 181–209 (2008)
E. DiBenedetto, U. Gianazza, V. Vespri, Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 9, 385–422 (2010)
E. DiBenedetto, U. Gianazza, V. Vespri, Harnack estimates and Hölder continuity for solutions to singular parabolic partial differential equations in the sub-critical range. Manuscripta Math. 131, 231–245 (2010)
E. DiBenedetto, U. Gianazza, V. Vespri, A new approach to the expansion of positivity set of non-negative solutions to certain singular parabolic partial differential equations. Proc. Am. Math. Soc. 138, 3521–3529 (2010)
E. DiBenedetto, U. Gianazza, V. Vespri, Liouville-type theorems for certain degenerate and singular parabolic equations. C. R. Acad. Sci. Paris Ser. I 348, 873–877 (2010)
E. DiBenedetto, U. Gianazza, V. Vespri, Harnack’s Inequality for Degenerate and Singular Parabolic Equations. Springer Monographs in Mathematics (Springer, New York/Heidelberg, 2012)
F. Duzaar, G. Mingione, K. Steffen, Parabolic systems with polynomial growth and regularity. Mem. Am. Math. Soc. 214(1005), x+118 pp. (2011)
E.B. Fabes, N. Garofalo, Parabolic B.M.O. and Harnack’s inequality. Proc. Am. Math. Soc. 50, 63–69 (1985)
E.B. Fabes, D.W. Stroock, A new proof of Moser’s parabolic Harnack inequality via the old ideas of Nash. Arch. Rat. Mech. Anal. 96, 327–338 (1986)
A. Farina, A Bernstein-type result for the minimal surface equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. 14, 1231–1237 (2015)
E. Ferretti, M.V. Safonov, Growth theorems and Harnack inequality for second order parabolic equations, in Harmonic Analysis and Boundary Value Problems (Fayetteville, AR, 2000). Contemporary Mathematics, vol. 277 (American Mathematical Society, Providence, 2001), pp. 87–112
S. Fornaro, M. Sosio, Intrinsic Harnack estimates for some doubly nonlinear degenerate parabolic equations. Adv. Differ. Equ. 13, 139–168 (2008)
S. Fornaro, V. Vespri, Harnack estimates for non negative weak solutions of singular parabolic equations satisfying the comparison principle. Manuscrpita Math. 141, 85–103 (2013)
S. Fornaro, M. Sosio, V. Vespri, Harnack type inequalities for some doubly nonlinear singular parabolic equations. Discrete Contin. Dyn. Syst. Ser. A 35, 5909–5926 (2015)
U. Gianazza, V. Vespri, Parabolic De Giorgi classes of order p and the Harnack inequality. Calc. Var. Partial Differ. Equ. 26, 379–399 (2006)
U. Gianazza, V. Vespri, A Harnack inequality for solutions of doubly nonlinear parabolic equations. J. Appl. Funct. Anal. 1, 271–284 (2006)
U. Gianazza, M. Surnachev, V. Vespri, On a new proof of Hölder continuity of solutions of p-Laplace type parabolic equations. Adv. Calc. Var. 3, 263–278 (2010)
E. Giusti, Direct Methods in the Calculus of Variations (World Scientific Publishing Co., Inc., River Edge, 2003)
A. Grigor’yan, The heat equation on non-compact Riemannian manifolds. Matem. Sbornik 182, 55–87 (1991). Engl. transl. Math. USSR Sb. 72, 47–77 (1992)
J. Hadamard, Extension à l’ équation de la chaleur d’ un theoreme de A. Harnack. Rend. Circ. Mat. Palermo 3, 337–346 (1954)
R.S. Hamilton, A matrix Harnack estimate for the heat equation. Commun. Anal. Geom. 1, 113–126 (1993)
C.G.A. von Harnack, Die Grundlagen der Theorie des logaritmischen Potentiales und der eindeutigen Potentialfunktion in der Ebene (Teubner, Leipzig, 1887)
M.A. Herrero, M. Pierre, The Cauchy problem for u t = Δ(u m) when 0 < m < 1. Trans. Am. Math. Soc. 291, 145–158 (1985)
C. Imbert, L. Silvestre, An introduction to fully nonlinear parabolic equations, in An Introduction to the Kḧler-Ricci Flow. Lecture Notes in Mathematics vol. 2086 (Springer, Cham, 2013), pp. 7–88
A.V. Ivanov, Regularity for doubly nonlinear parabolic equations. J. Math. Sci. 83, 22 (1997)
F. John, L. Nirenberg, On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415–426 (1961)
A.S. Kalashnikov, Some problems of the qualitative theory of nonlinear degenerate second order parabolic equations. Russ. Math. Surv. 42, 169–222 (1987)
S. Kamin, J.L. Vázquez, Fundamental solutions and asymptotic behavior for the p-Laplacian equation. Rev. Mat. Iberoamericana 4, 339–354 (1988)
M. Kassmann, Harnack inequalities: an introduction. Bound. Value Probl. 2007, 81415 (2007)
J. Kinnunen, Regularity for a doubly nonlinear parabolic equation, in Geometric Aspects of Partial Differential Equations. RIMS Kôkyûroku, vol. 1842 (Kyoto University, 2013), pp. 40–60
J. Kinnunen, T. Kuusi, Local behavior of solutions to doubly nonlinear parabolic equations. Math. Ann. 337, 705–728 (2007)
N.V. Krylov, Fully nonlinear second order elliptic equations: recent development. Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. Ser. 4 25, 569–595 (1997)
N.V. Krylov, M.V. Safonov, A certain property of solutions of parabolic equations with measurable coefficients. Izv. Akad. Nauk SSSR Ser. Mat. 44, 161–175 (1980) (in Russian). Translated in Math. of the USSR Izv. 16, 151–164 (1981)
B.L. Kotschwar, Hamilton’s gradient estimate for the heat kernel on complete manifolds. Proc. Am. Math. Soc. 135, 3013–3019 (2007)
M. Küntz, P. Lavallée, Experimental evidence and theoretical analysis of anomalous diffusion during water infiltration in porous building materials. J. Phys. D: Appl. Phys. 34, 2547–2554 (2001)
T. Kuusi, G. Mingione, Nonlinear potential theory of elliptic systems. Nonlinear Anal. 138, 277–299 (2016)
O.A. Ladyzenskaya, N.A. Solonnikov, N.N. Uraltzeva, Linear and quasilinear equations of parabolic type. Translations of Mathematical Monographs, vol. 23 (American Mathematical Society, Providence, RI, 1967)
E.M. Landis, Second Order Equations of Elliptic and Parabolic Type. Translations of Mathematical Monographs, vol. 171 (American Mathematical Society, Providence, RI, 1998; Nauka, Moscow, 1971)
P. Li, Harmonic functions on complete Riemannian manifolds, in Handbook of Geometric Analysis, vol. I. Advanced Lectures in Mathematics, vol. 7 (Higher Education Press and International Press, Beijing/Boston, 2008), pp. 195–227
P. Li, S.T. Yau, On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986)
F. Lin, Q.S. Zhang, On ancient solutions of the heat equation. Arxiv preprint. arXiv:1712.04091v2
D. Maldonado, On the elliptic Harnack inequality. Proc. Am. Math. Soc. 145, 3981–3987 (2017)
J.H. Michael, L.M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of \(\mathbb {R}^{N}\). Commun. Pure Appl. Math. 26, 361–379 (1973)
G. Mingione, Regularity of minima: an invitation to the Dark Side of the Calculus of Variations. Appl. Math. 51, 355–426 (2006)
J. Moser, On Harnack’s theorem for elliptic differential equations. Commun. Pure Appl. Math. 14, 577–591 (1961)
J. Moser, A Harnack inequality for parabolic differential equations. Commun. Pure Appl. Math. 17, 101–134 (1964)
J. Moser, On a pointwise estimate for parabolic differential equations. Commun. Pure Appl. Math. 24, 727–740 (1971)
R. Müller, Differential Harnack Inequalities and the Ricci Flow. EMS Series of Lectures in Mathematics (European Mathematical Society (EMS), Zürich, 2006)
J. Nash, Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)
B. Pini, Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico. Rend. Sem. Mat. Univ. Padova 23, 422–434 (1954)
M.M. Porzio, V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations. J. Differ. Equ. 103, 146–178 (1993)
F. Ragnedda, S. Vernier Piro, V. Vespri, Pointwise estimates for the fundamental solutions of a class of singular parabolic problems. J. Anal. Math. 121, 235–253 (2013)
L. Saloff-Coste, A note on Poincare, Sobolev and Harnack inequalities. Duke Math. J. 65, 27–38 (1992)
L. Saloff-Coste, Uniformly elliptic operators on Riemannian manifolds. J. Differ. Geom. 36, 417–450 (1992)
L. Saloff-Coste, Aspects of Sobolev-Type Inequalities. London Mathematical Society Lecture Notes Series, vol. 289 (Cambridge University Press, Cambridge, 2001)
J. Serrin, Local behavior of solutions of quasi-linear equations. Acta Math. 111, 247–302 (1964)
R.E. Showalter, N.J. Walkington, Diffusion of fluid in a fissured medium with micro-stricture. SIAM J. Mat. Anal. 22, 1702–1722 (1991)
Ph. Souplet, Q.S. Zhang, Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds. Bull. Lond. Math. Soc. 38, 1045–1053 (2006)
A.F. Tedeev, V. Vespri, Optimal behavior of the support of the solutions to a class of degenerate parabolic systems. Interfaces Free Bound. 17, 143–156 (2015)
E.V. Teixeira, J.M. Urbano, An intrinsic Liouville theorem for degenerate parabolic equations. Arch. Math. 102, 483–487 (2014)
N.S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic partial differential equations. Commun. Pure Appl. Math. 20, 721–747 (1967)
N.S. Trudinger, Pointwise estimates and quasilinear parabolic equations. Commun. Pure Appl. Math. 21, 205–226 (1968)
N.S. Trudinger, A new proof of the interior gradient bound for the minimal surface equation in N dimensions. Proc. Nat. Acad. Sci. U.S.A. 69, 821–823 (1972)
K. Uhlenbeck, Regularity for a class of nonlinear elliptic systems. Acta Math. 138, 219–240 (1977)
J.M. Urbano, The Method of Intrinsic Scaling. Lecture Notes in Mathematics, vol. 1930 (Springer, Berlin, 2008)
J.L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type. Oxford Lecture Series in Mathematics and its Applications, vol. 33 (Oxford University Press, Oxford, 2006)
J.L. Vázquez, The Porous Medium Equation: Mathematical Theory. Oxford Mathematical Monographs (Oxford Science Publications, Clarendon Press, Oxford, 2012)
D.V. Widder, The role of the Appell transformation in the theory of heat conduction. Trans. Am. Math. Soc. 109, 121–134 (1963)
S.T. Yau, Harmonic functions on complete Riemannian manifolds. Commun. Pure Appl. Math. 28, 201–228 (1975)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Düzgün, F.G., Mosconi, S., Vespri, V. (2019). Harnack and Pointwise Estimates for Degenerate or Singular Parabolic Equations. In: Dipierro, S. (eds) Contemporary Research in Elliptic PDEs and Related Topics. Springer INdAM Series, vol 33. Springer, Cham. https://doi.org/10.1007/978-3-030-18921-1_8
Download citation
DOI: https://doi.org/10.1007/978-3-030-18921-1_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-18920-4
Online ISBN: 978-3-030-18921-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)