Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations
DiBenedetto, Emmanuele1 ; Gianazza, Ugo2 ; Vespri, Vincenzo3
1 Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville TN 37240, USA
2 Dipartimento di Matematica “F. Casorati”, Università di Pavia, via Ferrata, 1, 27100 Pavia, Italia
3 Dipartimento di Matematica “U. Dini”, Università di Firenze, viale Morgagni, 67/A, 50134 Firenze, Italia
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 2, pp. 385-422.

Forward, backward and elliptic Harnack inequalities for non-negative solutions of a class of singular, quasi-linear, parabolic equations, are established. These classes of singular equations include the p-Laplacean equation and equations of the porous medium type. Key novel points include form of a Harnack estimate backward in time, that has never been observed before, and measure theoretical proofs, as opposed to comparison principles. These Harnack estimates are established in the super-critical range (1.5) below. Such a range is optimal for a Harnack estimate to hold.

Classification : 35K65, 35B65, 35B45
DiBenedetto, Emmanuele 1 ; Gianazza, Ugo 2 ; Vespri, Vincenzo 3

1 Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville TN 37240, USA
2 Dipartimento di Matematica “F. Casorati”, Università di Pavia, via Ferrata, 1, 27100 Pavia, Italia
3 Dipartimento di Matematica “U. Dini”, Università di Firenze, viale Morgagni, 67/A, 50134 Firenze, Italia 
@article{ASNSP_2010_5_9_2_385_0,
     author = {DiBenedetto, Emmanuele and Gianazza, Ugo and Vespri, Vincenzo},
     title = {Forward, backward and elliptic {Harnack} inequalities for non-negative solutions to certain singular parabolic partial differential equations},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {385--422},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 9},
     number = {2},
     year = {2010},
     mrnumber = {2731161},
     zbl = {1206.35053},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2010_5_9_2_385_0/}
}
TY  - JOUR
AU  - DiBenedetto, Emmanuele
AU  - Gianazza, Ugo
AU  - Vespri, Vincenzo
TI  - Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2010
SP  - 385
EP  - 422
VL  - 9
IS  - 2
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2010_5_9_2_385_0/
LA  - en
ID  - ASNSP_2010_5_9_2_385_0
ER  - 
%0 Journal Article
%A DiBenedetto, Emmanuele
%A Gianazza, Ugo
%A Vespri, Vincenzo
%T Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2010
%P 385-422
%V 9
%N 2
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2010_5_9_2_385_0/
%G en
%F ASNSP_2010_5_9_2_385_0
DiBenedetto, Emmanuele; Gianazza, Ugo; Vespri, Vincenzo. Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 2, pp. 385-422. http://archive.numdam.org/item/ASNSP_2010_5_9_2_385_0/

[1] Y. Z. Chen and E. Dibenedetto, Hölder estimates of solutions of singular parabolic equations with measurable coefficients, Arch. Ration. Mech. Anal. 118 (1992), 257–271. | MR | Zbl

[2] 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. | MR | Zbl

[3] E. Dibenedetto, Intrinsic Harnack type inequalities for solutions of certain degenerate parabolic equations, Arch. Ration. Mech. Anal. 100 (1988), 129–147. | MR | Zbl

[4] E. Dibenedetto and Y. C. Kwong, Intrinsic Harnack estimates and extinction profile for certain singular parabolic equations, Trans. Amer. Math. Soc. 330 (1992), 783–811. | Zbl

[5] E. Dibenedetto, Y. C. Kwong and V. Vespri, Local space analiticity of solutions of certain singular parabolic equations, Indiana Univ. Math. J. 40 (1991), 741–765. | MR | Zbl

[6] E. Dibenedetto, “Degenerate Parabolic Equations”, Springer Verlag, Series Universitext, New York, 1993. | MR | Zbl

[7] E. Dibenedetto, U. Gianazza and V. Vespri, Harnack estimates for quasi-linear degenerate parabolic differential equations, Acta Math. 200 (2008), 181–209. | MR | Zbl

[8] J. Hadamard, Extension à l’équation de la chaleur d’un théorème de A. Harnack, Rend. Circ. Mat. Palermo 3 (1954), 337–346. | MR | Zbl

[9] B. H. Gilding, On a class of similarity solutions of the porous media equation III, J. Math. Anal. Appl. 77 (1980), 381–402. | MR | Zbl

[10] J. Moser, On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591. | MR | Zbl

[11] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101–134. | MR | Zbl

[12] J. Moser, On a pointwise estimate for parabolic differential equations, Comm. Pure Appl. Math. 24 (1971), 727–740. | MR | Zbl

[13] M. A. Peletier and H. Zhang, Self-similar solutions of a fast diffusion equation that do not conserve mass, Differential Integral Equations 8 (1995), 2045–2064. | MR | Zbl

[14] 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. | EuDML | Numdam | MR | Zbl