Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Harnack-type inequalities for evolution equations

Authors: Giles Auchmuty and David Bao
Journal: Proc. Amer. Math. Soc. 122 (1994), 117-129
MSC: Primary 35K22; Secondary 35B45, 35Q40
MathSciNet review: 1219716
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Here we derive Harnack inequalities for nonnegative solutions of the porous medium equation and the p-diffusion equation. The method applies to functions obeying certain a priori evolution inequalities. The proofs are based on optimizing inequalities for the convective derivative of the function along a path.

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

  • [1] D. G. Aronson, The porous medium equation, Nonlinear diffusion problems (Montecatini Terme, 1985) Lecture Notes in Math., vol. 1224, Springer, Berlin, 1986, pp. 1–46. MR 877986, 10.1007/BFb0072687
  • [2] Donald G. Aronson and Philippe Bénilan, Régularité des solutions de l’équation des milieux poreux dans 𝑅^{𝑁}, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 2, A103–A105 (French, with English summary). MR 524760
  • [3] E. DiBenedetto, Topics in quasilinear degenerate and singular parabolic equations, Vorlesungsreihe no. 20, SFB 256, Universität Bonn, 1991.
  • [4] Emmanuele DiBenedetto and Avner Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1–22. MR 783531, 10.1515/crll.1985.357.1
  • [5] J. R. Esteban and J. L. Vásquez, Regularity of nonnegative solutions of the p-Laplacian parabolic equation, $ {1^a}$ Reunion Hispano-Italiana de Analisis No Lineal y Matematica Aplicada, 1989, pp. 87-92.
  • [6] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
  • [7] R. Hamilton, Lecture given at the NSF Summer Institute on Differential Geometry, UCLA, 1990.
  • [8] A. Harnack, Grundlagen der Theorie des Logarithmischen Potentials, Teubner, Leipzig, 1887.
  • [9] Peter Li and Shing-Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153–201. MR 834612, 10.1007/BF02399203
  • [10] Jürgen Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101–134. MR 0159139
  • [11] M. Marcus and V. J. Mizel, Absolute continuity on tracks and mappings of Sobolev spaces, Arch. Rational Mech. Anal. 45 (1972), 294–320. MR 0338765

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35K22, 35B45, 35Q40

Retrieve articles in all journals with MSC: 35K22, 35B45, 35Q40

Additional Information

Article copyright: © Copyright 1994 American Mathematical Society