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)

Request Permissions   Purchase Content 
 
 

 

Lower semicontinuity of weak supersolutions to the porous medium equation


Authors: Benny Avelin and Teemu Lukkari
Journal: Proc. Amer. Math. Soc. 143 (2015), 3475-3486
MSC (2010): Primary 35K55, 31C45
DOI: https://doi.org/10.1090/proc/12727
Published electronically: April 23, 2015
MathSciNet review: 3348790
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Weak supersolutions to the porous medium equation are defined by means of smooth test functions under an integral sign. We show that nonnegative weak supersolutions become lower semicontinuous after redefinition on a set of measure zero. This shows that weak supersolutions belong to a class of supersolutions defined by a comparison principle.


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

  • [1] G. I. Barenblatt, On self-similar motions of a compressible fluid in a porous medium, Akad. Nauk SSSR. Prikl. Mat. Meh. 16 (1952), 679-698 (Russian). MR 0052948 (14,699h)
  • [2] Björn E. J. Dahlberg and Carlos E. Kenig, Nonnegative solutions of the porous medium equation, Comm. Partial Differential Equations 9 (1984), no. 5, 409-437. MR 741215 (85j:35099), https://doi.org/10.1080/03605308408820336
  • [3] Panagiota Daskalopoulos and Carlos E. Kenig, Degenerate diffusions, Initial value problems and local regularity theory. EMS Tracts in Mathematics, vol. 1, European Mathematical Society (EMS), Zürich, 2007. MR 2338118 (2009b:35214)
  • [4] Emmanuele DiBenedetto, Degenerate parabolic equations, Universitext, Springer-Verlag, New York, 1993. MR 1230384 (94h:35130)
  • [5] Emmanuele DiBenedetto and Avner Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1-22. MR 783531 (87f:35134a), https://doi.org/10.1515/crll.1985.357.1
  • [6] Juha Kinnunen and Peter Lindqvist, Definition and properties of supersolutions to the porous medium equation, J. Reine Angew. Math. 618 (2008), 135-168. MR 2404748 (2009c:35250), https://doi.org/10.1515/CRELLE.2008.035
  • [7] Tuomo Kuusi, Lower semicontinuity of weak supersolutions to nonlinear parabolic equations, Differential Integral Equations 22 (2009), no. 11-12, 1211-1222. MR 2555645 (2010i:35214)
  • [8] Joachim Naumann, Einführung in die Theorie parabolischer Variationsungleichungen, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 64, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1984 (German). With English, French and Russian summaries. MR 771666 (86e:35080)
  • [9] Frédéric Riesz, Sur les Fonctions Subharmoniques et Leur Rapport à la Théorie du Potentiel, Acta Math. 48 (1926), no. 3-4, 329-343 (French). MR 1555229, https://doi.org/10.1007/BF02565338
  • [10] Juan Luis Vázquez, The porous medium equation, Mathematical theory. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. MR 2286292 (2008e:35003)
  • [11] Zhuoqun Wu, Junning Zhao, Jingxue Yin, and Huilai Li, Nonlinear diffusion equations, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. Translated from the 1996 Chinese original and revised by the authors. MR 1881297 (2003e:35002)
  • [12] Ya. B. Zeldovič and A. S. Kompaneec, On the theory of propagation of heat with the heat conductivity depending upon the temperature, Collection in honor of the seventieth birthday of academician A. F. Ioffe, Izdat. Akad. Nauk SSSR, Moscow, 1950, pp. 61-71. MR 0069381 (16,1029c)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35K55, 31C45

Retrieve articles in all journals with MSC (2010): 35K55, 31C45


Additional Information

Benny Avelin
Affiliation: Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden
Email: benny.avelin@math.uu.se

Teemu Lukkari
Affiliation: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, 40014 Jyväskylä, Finland
Email: teemu.j.lukkari@jyu.fi

DOI: https://doi.org/10.1090/proc/12727
Keywords: Porous medium equation, supersolutions, comparison principle, lower semicontinuity, degenerate diffusion
Received by editor(s): December 10, 2013
Published electronically: April 23, 2015
Additional Notes: The research reported in this work was done during the authors’ stay at the Institut Mittag-Leffler (Djursholm, Sweden).
Communicated by: Joachim Krieger
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society