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
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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.


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  • [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

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1219716-X
Article copyright: © Copyright 1994 American Mathematical Society