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Harnack and Pointwise Estimates for Degenerate or Singular Parabolic Equations

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Contemporary Research in Elliptic PDEs and Related Topics

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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.

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Notes

  1. 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. 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

  1. D.G. Aronson, Bounds for the fundamental solution of a parabolic equation. Bull. Am. Math. Soc. 73, 890–896 (1967)

    Article  MathSciNet  Google Scholar 

  2. D.G. Aronson, J. Serrin, Local behavior of solutions of quasilinear parabolic equations. Arch. Ration. Mech. Anal. 25, 81–122 (1967)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  4. G.I. Barenblatt, A.S. Monin, Flying sources and the microstructure of the ocean: a mathematical theory. Uspekhi Mat. Nauk. 37, 125–126 (1982)

    MathSciNet  Google Scholar 

  5. G.I. Barenblatt, V.M. Entov, V.M. Rizhnik, Motion of Fluids and Gases in Natural Strata (Nedra, Moscow, 1984)

    Google Scholar 

  6. M. Barlow, M. Murugan, Stability of the elliptic Harnack inequality. Ann. Math. 187, 777–823 (2018)

    Article  MathSciNet  Google Scholar 

  7. 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)

    Google Scholar 

  8. V. Bögelein, F. Ragnedda, S. Vernier Piro, V. Vespri, Moser-Nash kernel estimates for degenerate parabolic equations. J. Funct. Anal. 272, 2956–2986 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. 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)

    Article  MathSciNet  Google Scholar 

  13. 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)

    Article  MathSciNet  Google Scholar 

  14. 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)

    Article  MathSciNet  Google Scholar 

  15. Y.Z. Chen, E. DiBenedetto, On the local behaviour of solutions of singular parabolic equations. Arch. Ration. Mech. Anal. 103, 319–346 (1988)

    Article  Google Scholar 

  16. S.Y. Cheng, S.T. Yau, Differential equations on Riemannian manifolds and their geometric applications. Commun. Pure Appl. Math. 28, 333–354 (1975)

    Article  MathSciNet  Google Scholar 

  17. T.H. Colding, W.P. Minicozzi II, Harmonic functions on manifolds. Ann. Math. 146, 725–747 (1997)

    Article  MathSciNet  Google Scholar 

  18. 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)

    MathSciNet  MATH  Google Scholar 

  19. 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)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. E. DiBenedetto, Degenerate Parabolic Equations. Universitext (Springer, New York, 1993)

    Book  Google Scholar 

  22. E. DiBenedetto, A. Friedman, Hölder estimates for non-linear degenerate parabolic systems. J. Reine Angew. Math. 357, 1–22 (1985)

    MathSciNet  MATH  Google Scholar 

  23. E. DiBenedetto, U. Gianazza, Some properties of De Giorgi classes. Rend. Istit. Mat. Univ. Trieste 48, 245–260 (2016)

    MathSciNet  MATH  Google Scholar 

  24. E. DiBenedetto, Y.C. Kwong, Intrinsic Harnack estimates and extinction profile for certain singular parabolic equations. Trans. Am. Math. Soc. 330, 783–811 (1992)

    Article  Google Scholar 

  25. E. DiBenedetto, N.S. Trudinger, Harnack inequalities for quasi-minima of Variational integrals. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 295–308 (1984)

    Article  MathSciNet  Google Scholar 

  26. E. DiBenedetto, U. Gianazza, V. Vespri, Harnack Estimates for quasi-linear degenerate parabolic differential equation. Acta Math. 200, 181–209 (2008)

    Article  MathSciNet  Google Scholar 

  27. 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)

    Google Scholar 

  28. 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)

    Article  MathSciNet  Google Scholar 

  29. 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)

    Article  MathSciNet  Google Scholar 

  30. 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)

    Article  MathSciNet  Google Scholar 

  31. E. DiBenedetto, U. Gianazza, V. Vespri, Harnack’s Inequality for Degenerate and Singular Parabolic Equations. Springer Monographs in Mathematics (Springer, New York/Heidelberg, 2012)

    Google Scholar 

  32. F. Duzaar, G. Mingione, K. Steffen, Parabolic systems with polynomial growth and regularity. Mem. Am. Math. Soc. 214(1005), x+118 pp. (2011)

    Article  MathSciNet  Google Scholar 

  33. E.B. Fabes, N. Garofalo, Parabolic B.M.O. and Harnack’s inequality. Proc. Am. Math. Soc. 50, 63–69 (1985)

    Google Scholar 

  34. 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)

    Article  Google Scholar 

  35. A. Farina, A Bernstein-type result for the minimal surface equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. 14, 1231–1237 (2015)

    MathSciNet  MATH  Google Scholar 

  36. 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

    Google Scholar 

  37. S. Fornaro, M. Sosio, Intrinsic Harnack estimates for some doubly nonlinear degenerate parabolic equations. Adv. Differ. Equ. 13, 139–168 (2008)

    MathSciNet  MATH  Google Scholar 

  38. 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)

    Article  Google Scholar 

  39. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  41. U. Gianazza, V. Vespri, A Harnack inequality for solutions of doubly nonlinear parabolic equations. J. Appl. Funct. Anal. 1, 271–284 (2006)

    MathSciNet  MATH  Google Scholar 

  42. 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)

    Article  MathSciNet  Google Scholar 

  43. E. Giusti, Direct Methods in the Calculus of Variations (World Scientific Publishing Co., Inc., River Edge, 2003)

    Book  Google Scholar 

  44. 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)

    Google Scholar 

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

    Article  Google Scholar 

  46. R.S. Hamilton, A matrix Harnack estimate for the heat equation. Commun. Anal. Geom. 1, 113–126 (1993)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  48. 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)

    MathSciNet  MATH  Google Scholar 

  49. 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

    Google Scholar 

  50. A.V. Ivanov, Regularity for doubly nonlinear parabolic equations. J. Math. Sci. 83, 22 (1997)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  52. A.S. Kalashnikov, Some problems of the qualitative theory of nonlinear degenerate second order parabolic equations. Russ. Math. Surv. 42, 169–222 (1987)

    Article  Google Scholar 

  53. S. Kamin, J.L. Vázquez, Fundamental solutions and asymptotic behavior for the p-Laplacian equation. Rev. Mat. Iberoamericana 4, 339–354 (1988)

    Article  MathSciNet  Google Scholar 

  54. M. Kassmann, Harnack inequalities: an introduction. Bound. Value Probl. 2007, 81415 (2007)

    Article  MathSciNet  Google Scholar 

  55. 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

    Google Scholar 

  56. J. Kinnunen, T. Kuusi, Local behavior of solutions to doubly nonlinear parabolic equations. Math. Ann. 337, 705–728 (2007)

    Article  MathSciNet  Google Scholar 

  57. 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)

    MathSciNet  MATH  Google Scholar 

  58. 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)

    Google Scholar 

  59. B.L. Kotschwar, Hamilton’s gradient estimate for the heat kernel on complete manifolds. Proc. Am. Math. Soc. 135, 3013–3019 (2007)

    Article  MathSciNet  Google Scholar 

  60. 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)

    Article  Google Scholar 

  61. T. Kuusi, G. Mingione, Nonlinear potential theory of elliptic systems. Nonlinear Anal. 138, 277–299 (2016)

    Article  MathSciNet  Google Scholar 

  62. 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)

    Google Scholar 

  63. 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)

    Google Scholar 

  64. 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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  66. F. Lin, Q.S. Zhang, On ancient solutions of the heat equation. Arxiv preprint. arXiv:1712.04091v2

    Google Scholar 

  67. D. Maldonado, On the elliptic Harnack inequality. Proc. Am. Math. Soc. 145, 3981–3987 (2017)

    Article  MathSciNet  Google Scholar 

  68. 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)

    Google Scholar 

  69. G. Mingione, Regularity of minima: an invitation to the Dark Side of the Calculus of Variations. Appl. Math. 51, 355–426 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  71. J. Moser, A Harnack inequality for parabolic differential equations. Commun. Pure Appl. Math. 17, 101–134 (1964)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  73. R. Müller, Differential Harnack Inequalities and the Ricci Flow. EMS Series of Lectures in Mathematics (European Mathematical Society (EMS), Zürich, 2006)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  75. 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)

    MathSciNet  MATH  Google Scholar 

  76. M.M. Porzio, V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations. J. Differ. Equ. 103, 146–178 (1993)

    Article  Google Scholar 

  77. 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)

    Article  MathSciNet  Google Scholar 

  78. L. Saloff-Coste, A note on Poincare, Sobolev and Harnack inequalities. Duke Math. J. 65, 27–38 (1992)

    MATH  Google Scholar 

  79. L. Saloff-Coste, Uniformly elliptic operators on Riemannian manifolds. J. Differ. Geom. 36, 417–450 (1992)

    Article  MathSciNet  Google Scholar 

  80. L. Saloff-Coste, Aspects of Sobolev-Type Inequalities. London Mathematical Society Lecture Notes Series, vol. 289 (Cambridge University Press, Cambridge, 2001)

    Google Scholar 

  81. J. Serrin, Local behavior of solutions of quasi-linear equations. Acta Math. 111, 247–302 (1964)

    Article  MathSciNet  Google Scholar 

  82. R.E. Showalter, N.J. Walkington, Diffusion of fluid in a fissured medium with micro-stricture. SIAM J. Mat. Anal. 22, 1702–1722 (1991)

    Article  Google Scholar 

  83. 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)

    Article  MathSciNet  Google Scholar 

  84. 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)

    Article  MathSciNet  Google Scholar 

  85. E.V. Teixeira, J.M. Urbano, An intrinsic Liouville theorem for degenerate parabolic equations. Arch. Math. 102, 483–487 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  87. N.S. Trudinger, Pointwise estimates and quasilinear parabolic equations. Commun. Pure Appl. Math. 21, 205–226 (1968)

    Article  MathSciNet  Google Scholar 

  88. 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)

    Article  MathSciNet  Google Scholar 

  89. K. Uhlenbeck, Regularity for a class of nonlinear elliptic systems. Acta Math. 138, 219–240 (1977)

    Article  MathSciNet  Google Scholar 

  90. J.M. Urbano, The Method of Intrinsic Scaling. Lecture Notes in Mathematics, vol. 1930 (Springer, Berlin, 2008)

    Google Scholar 

  91. 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)

    Google Scholar 

  92. J.L. Vázquez, The Porous Medium Equation: Mathematical Theory. Oxford Mathematical Monographs (Oxford Science Publications, Clarendon Press, Oxford, 2012)

    Google Scholar 

  93. D.V. Widder, The role of the Appell transformation in the theory of heat conduction. Trans. Am. Math. Soc. 109, 121–134 (1963)

    Article  MathSciNet  Google Scholar 

  94. S.T. Yau, Harmonic functions on complete Riemannian manifolds. Commun. Pure Appl. Math. 28, 201–228 (1975)

    Article  MathSciNet  Google Scholar 

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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.

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  1. Department of Mathematics, Hacettepe University, Ankara, Turkey

    Fatma Gamze Düzgün

  2. Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Catania, Italy

    Sunra Mosconi

  3. Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, Firenze, Italy

    Vincenzo Vespri

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    Serena Dipierro

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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

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