In this work we prove both local and global Harnack estimates for weak supersolutions to second order nonlinear degenerate parabolic partial differential equations in divergence form. We reduce the proof to an analysis of so-called hot and cold alternatives, and use the expansion of positivity together with a parabolic type of covering argument. Our proof uses only the properties of weak supersolutions. In particular, no comparison to weak solutions is needed.
@article{ASNSP_2008_5_7_4_673_0,
author = {Kuusi, Tuomo},
title = {Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {673--716},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 7},
number = {4},
year = {2008},
zbl = {1178.35100},
mrnumber = {2483640},
language = {en},
url = {http://archive.numdam.org/item/ASNSP_2008_5_7_4_673_0/}
}
TY - JOUR AU - Kuusi, Tuomo TI - Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2008 DA - 2008/// SP - 673 EP - 716 VL - Ser. 5, 7 IS - 4 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2008_5_7_4_673_0/ UR - https://zbmath.org/?q=an%3A1178.35100 UR - https://www.ams.org/mathscinet-getitem?mr=2483640 LA - en ID - ASNSP_2008_5_7_4_673_0 ER -
Kuusi, Tuomo. Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 7 (2008) no. 4, pp. 673-716. http://archive.numdam.org/item/ASNSP_2008_5_7_4_673_0/
[1] and , Gradient estimates for a class of parabolic systems, Duke Math. J. 136 (2) (2007), 285-320. | MR 2286632 | Zbl 1113.35105
[2] and , The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc. 280 (1) (1983), 351-366. | MR 712265 | Zbl 0556.76084
[3] and , Local behavior of solutions of quasilinear parabolic equations, Arch. Ration. Mech. Anal. 25 (1967), 81-122. | MR 244638 | Zbl 0154.12001
[4] and , Cauchy problem for nonlinear parabolic equations, Hokkaido Math. J. 27 (1) (1998), 51-75. | MR 1608612 | Zbl 0902.35060
[5] and , Non-Negative solutions of the porous medium equation, Comm. Partial Differential Equations 9 (5) (1984), 409-437. | MR 741215 | Zbl 0547.35057
[6] and , “Degenerate Diffusions", EMS Tracts in Mathematics, Vol. 1, European Mathematical Society (EMS), Zürich, 2007, Initial value problems and local regularity theory. | MR 2338118 | Zbl 1205.35002
[7] , Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Natur. 3 (3) (1957), 25-43. | MR 93649 | Zbl 0084.31901
[8] , “Degenerate Parabolic Equations", Universitext, Springer-Verlag, New York, 1993. | MR 1230384 | Zbl 0794.35090
[9] , and , Intrinsic Harnack estimates for non-negative local solutions of degenerate parabolic equations, Acta Math. 200 (2008), 181-209. | MR 2413134 | Zbl 1221.35213
[10] , and , Subpotential lower bounds for nonnegative solutions to certain quasi-linear degenerate parabolic equations, Duke Math. J. 143 (2008), 1-15. | MR 2414742 | Zbl 1170.35054
[11] and , On the Cauchy problem and initial traces for a degenerate parabolic equation, Trans. Amer. Math. Soc. 314 (1) (1989), 187-224. | MR 962278 | Zbl 0691.35047
[12] , and , Current issues on singular and degenerate evolution equations, In: “Evolutionary Equations", Vol. I, Handb. Differ. Equ., 169-286, North-Holland, Amsterdam, 2004. | MR 2103698 | Zbl 1082.35002
[13] and , Harnack inequalities for quasiminima of variational integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 295-308, MR MR778976 (86g:49007). | Numdam | MR 778976 | Zbl 0565.35012
[14] and , A Harnack inequality for a degenerate parabolic equation, J. Evol. Equ. 6 (2) (2006), 247-267. | MR 2227696 | Zbl 1112.35113
[15] , “Introduction to Regularity Theory for Nonlinear Elliptic Systems", Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1993. | MR 1239172 | Zbl 0786.35001
[16] , “Lectures on Analysis on Metric Spaces", Universitext, Springer-Verlag, New York, 2001. | MR 1800917 | Zbl 0985.46008
[17] , and , “Nonlinear Potential Theory of Degenerate Elliptic Equations", Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1993, Oxford Science Publications. | MR 1207810 | Zbl 1115.31001
[18] , The Harnack inequality for generalized solutions of second order quasilinear parabolic equations, Trudy Mat. Inst. Steklov. 102 (1967), 51-84. | MR 232096 | Zbl 0204.11404
[19] and , Regularity of quasi-minimizers on metric spaces, Manuscripta Math. 105 (3) (2001), 401-423. | MR 1856619 | Zbl 1006.49027
[20] and , A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR Izv. 16 (1) (1981), 151-164. | MR 563790 | Zbl 0464.35035
[21] , On a Harnack inequality for nonlinear parabolic equations, Publ. Res. Inst. Math. Sci. 3 (1967/1968), 211-241. | MR 229954 | Zbl 0162.42102
[22] , “Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires", Dunod, 1969. | Zbl 0189.40603
[23] , On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577-591. | MR 159138 | Zbl 0111.09302
[24] , A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101-134. | MR 159139 | Zbl 0149.06902
[25] , Correction to A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 20 (1967), 231-236. | MR 203268 | Zbl 0149.07001
[26] , On a pointwise estimate for parabolic differential equations, Comm. Pure Appl. Math. 24 (1971), 727-740. | MR 288405 | Zbl 0227.35016
[27] , Pointwise estimates and quasilinear parabolic equations, Comm. Pure Appl. Math. 21 (1968), 205-226. | MR 226168 | Zbl 0159.39303
[28] , “The Method of Intrinsic Scaling. a Systematic Approach to Regularity for Degenerate and Singular PDEs”, Lecture Notes in Mathematics, Vol. 1930, Springer, 2008. | MR 2451216 | Zbl 1158.35003
[29] , “The Porous Medium Equation Mathematical Theory", Oxford Mathematical Monographs, Clarendon Press, 2006. | MR 2286292 | Zbl 1107.35003
[30] , “Smoothing and Decay Estimates for Nonlinear Diffusion Equations Equations of Porous Medium Type", Vol. 33 of Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, 2006. | MR 2282669 | Zbl 1113.35004
