On the viscosity solutions to Trudinger’s equation

  • Tilak Bhattacharya
  • Leonardo MarazziEmail author


We study the existence of positive viscosity solutions to Trudinger’s equation for cylindrical domains \({\Omega\times[0, T)}\), where \({\Omega\subset {I\!R}^{n}, n\ge 2,}\) is a bounded domain, T > 0 and \({2\le p < \infty}\). We show existence for general domains \({\Omega,}\) when \({n<p<\infty}\). For \({2\le p\le n}\), we prove existence for domains \({\Omega}\) that satisfy a uniform outer ball condition. We achieve this by constructing suitable sub-solutions and super-solutions and applying Perron’s method.

Mathematics Subject Classification

35K65 35K55 


  1. 1.
    Bhattacharya, T., DiBenedetto, E., Manfredi, J.J.: Limits as \({p\rightarrow \infty}\) of \({\Delta_pu_p=f}\) and related extremal problems. Some topics in nonlinear PDEs (Turin, 1989), Rend. Sem. Mat. Univ. Politec. Torino 1989, Special Issue, 15–68 (1991)Google Scholar
  2. 2.
    Bhattacharya, T., Marazzi, L.: On the viscosity solutions to a degenerate parabolic differential equation. Ann. Mat. Pura Appl. (June 4, 2014). doi: 10.1007/s10231-014-0427-1
  3. 3.
    Crandall M.G., Ishii H.: The maximum principle for semicontinuous Functions. Differ. Integr. Equ. 3(6), 1001–1014 (1990)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Crandall M.G., Ishii H., Lions P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1–67 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    DiBenedetto, E.: Degenerate Parabolic Equations. Universitext, Springer (1993)Google Scholar
  6. 6.
    Gianazza U., Vespri V.: A Harnack inequality for solutions of a doubly nonlinear parabolic equation. J. Appl. Funct. Anal. 1(3), 271–284 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Juutinen P., Lindqvist P.: Pointwise decay for the solutions of degenerate and singular parabolic equations. Adv. Differ. Equ. 14(7–8), 663–684 (2009)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Kinnunen J., Kuusi T.: Local behaviour of solutions to doubly nonlinear parabolic equations. Math. Ann. 337, 705–728 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Landis, E.M.: Second Order Equations of Elliptic and Parabolic Type. Translations of Mathematical Monographs (AMS) (1998)Google Scholar
  10. 10.
    Lindqvist P., Manfredi J.J.: Viscosity supersolutions of the evolutionary p-Laplace equation. Differ. Integr. Equ. 20(11), 1303–1319 (2007)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Orsina L., Porzio M.M.: \({L^{\infty}(Q)}\) -estimates and existence of solutions for some nonlinear parabolic equations. Bollettino U.M.I. 7(6-B), 631–647 (1992)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Porzio M.M.: On decay estimates. J. Evol. Equ. 9(3), 561–591 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Trudinger N.S.: Pointwise estimates and quasilinear parabolic equations. Commun. Pure Appl. Math. 21, 205–226 (1968)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of MathematicsWestern Kentucky UniversityBowling GreenUSA
  2. 2.Department of Liberal ArtsSavannah College of Arts and DesignSavannahUSA

Personalised recommendations