Electronic Research Announcements

ISSN 1079-6762

 

 

Intrinsic Harnack estimates for nonnegative local solutions of degenerate parabolic equations


Authors: Emmanuele DiBenedetto, Ugo Gianazza and Vincenzo Vespri
Journal: Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 95-99
MSC (2000): Primary 35K65, 35B65; Secondary 35B45
Published electronically: August 2, 2006
MathSciNet review: 2237273
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We establish the intrinsic Harnack inequality for nonnegative solutions of the parabolic $ p$-Laplacian equation by a proof that uses neither the comparison principle nor explicit self-similar solutions. The significance is that the proof applies to quasilinear $ p$-Laplacian-type equations, thereby solving a long-standing problem in the theory of degenerate parabolic equations.


References [Enhancements On Off] (What's this?)

  • 1. D. G. Aronson and James Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal. 25 (1967), 81–122. MR 0244638
  • 2. E. DiBenedetto, Harnack estimates in certain function classes, Atti Sem. Mat. Fis. Univ. Modena 37 (1989), no. 1, 173–182. MR 994063
  • 3. Emmanuele DiBenedetto, Degenerate parabolic equations, Universitext, Springer-Verlag, New York, 1993. MR 1230384
  • 4. E. DiBenedetto, U. Gianazza, and V. Vespri, Local clustering on the nonzero set of functions in $ W^{1,1}_{loc}(E)$, Rend. Lincei Mat. Appl. 17 (2006), 223-225.
  • 5. E. DiBenedetto, U. Gianazza, and V. Vespri, Harnack estimates for quasilinear degenerate parabolic differential equations, in preparation.
  • 6. J. Hadamard, Extension à l’équation de la chaleur d’un théorème de A. Harnack, Rend. Circ. Mat. Palermo (2) 3 (1954), 337–346 (1955) (French). MR 0068713
  • 7. O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, 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 (Russian). MR 0241822
  • 8. Bruno 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 (Italian). MR 0065794
  • 9. Jürgen Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101–134. MR 0159139
  • 10. Neil S. Trudinger, Pointwise estimates and quasilinear parabolic equations, Comm. Pure Appl. Math. 21 (1968), 205–226. MR 0226168

Similar Articles

Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (2000): 35K65, 35B65, 35B45

Retrieve articles in all journals with MSC (2000): 35K65, 35B65, 35B45


Additional Information

Emmanuele DiBenedetto
Affiliation: Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, USA
Email: em.diben@vanderbilt.edu

Ugo Gianazza
Affiliation: Dipartimento di Matematica “F. Casorati", Università di Pavia, via Ferrata 1, 27100 Pavia, Italy
Email: gianazza@imati.cnr.it

Vincenzo Vespri
Affiliation: Dipartimento di Matematica “U. Dini", Università di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy
Email: vespri@math.unifi.it

DOI: http://dx.doi.org/10.1090/S1079-6762-06-00166-1
Keywords: Degenerate parabolic equations, Harnack estimates, H\"older continuity
Received by editor(s): January 20, 2006
Published electronically: August 2, 2006
Communicated by: Luis A. Caffarelli
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.