Potential Analysis

, Volume 35, Issue 1, pp 1–38 | Cite as

Existence’s Results for Parabolic Problems Related to Fully Non Linear Operators Degenerate or Singular

  • Francoise Demengel


In this paper we prove some existence and regularity results concerning parabolic equations
$$ u_t = F(x, \nabla u, D^2 u) + f(x,t) $$
with some boundary conditions, on Ω×]0, T[, where Ω is some bounded domain which possesses the exterior cone property and F is some fully nonlinear elliptic operator, singular or degenerate.


Viscocity solutions Evolution problems Comparison principle 

Mathematics Subject Classifications (2010)

35J60 35J70 35K10 35K20 35K60 35K65 


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  1. 1.
    Barles, G., Perthame, B.: Discontinuous solutions of deterministic optimal stopping time problems. RAIRO Modél Math. Anal. Numér. 21, 557–579 (1987)MathSciNetMATHGoogle Scholar
  2. 2.
    Birindelli, I., Demengel, F.: Comparison principle and Liouville type results for singular fully nonlinear operators. Ann. Fac. Sci Toulouse Math 13, 261–287 (2004)MathSciNetMATHGoogle Scholar
  3. 3.
    Birindelli, I., Demengel, F.: First eigenvalue and maximum principle for fully nonlinear singular operators. Adv. Differ. Equ. 11(1), 91–119 (2006)MathSciNetMATHGoogle Scholar
  4. 4.
    Birindelli, I., Demengel, F.: Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Commun. Pure Appl. Anal. 6, 335–366 (2007)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Birindelli, I., Demengel, F.: The Dirichlet problem for singular fully nonlinear operators. Discrete Contin. Dyn. Syst. 110–121 (Special vol., 2007)Google Scholar
  6. 6.
    Birindelli, I., Demengel, F.: Eigenvalue and Dirichlet problem for fully-nonlinear operators in non smooth domains. J. Math. Anal. Appl. 352, 822–835 (2009)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Crandall, M., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27(1), 1–67 (1992)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Crandall, M.G., Kocan, M., Lions, P.L., Swiech, A.: Existence’s results for boundary problems for uniformly elliptic and parabolic fully non linear equation. Electr. J. Differ. Equ. 1999(N24) 1–20 (1999)MathSciNetGoogle Scholar
  9. 9.
    Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. I. J. Differ. Geom. 33(3), 635–681 (1991)MathSciNetMATHGoogle Scholar
  10. 10.
    Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. II. Trans. Am. Math. Soc. 330(1), 321–332 (1992)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. III. J. Geom. Anal. 2(n2), 121–150 (1992)MathSciNetMATHGoogle Scholar
  12. 12.
    Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. IV, I. J. Geom. Anal. 5(n1), 77–114 (1995)MathSciNetGoogle Scholar
  13. 13.
    Ishii, H.: Viscosity Solutions of Non-linear Partial Differential Equations. Sugaku Expositions, vol 9 (1996)Google Scholar
  14. 14.
    Ishii, H., Lions, P.L.: Viscosity solutions of fully- nonlinear second order elliptic partial differential equations. J. Differ. Equ. 83(1), 26–78 (1990)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Ishii, H., Souganidis, P.E.: Generalized motion of non compact hyperbolic surfaces with velocity haing arbitrary growth on the curvature tensor. Tohoku Math. J. 47, 227–250 (1995)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Juutinen, P., Kawhol, B.: On the evolution governed by the infinity Laplacian Math. Ann. 335(n4), 819–851 (2006)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Ohnuma, M., Sato, K.: Singular degenerate Parabolic equations with applications to the p-Laplace diffusion equation. Commun. Partial Differ. Equ. 22(3 and 4), 381–411 (1977)MathSciNetGoogle Scholar

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© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.University of Cergy-PontoiseCergy-PontoiseFrance

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