Minimizers of non convex scalar functionals and viscosity solutions of Hamilton-Jacobi equations


DOI: 10.1007/s00526-007-0124-7

Cite this article as:
Zagatti, S. Calc. Var. (2008) 31: 511. doi:10.1007/s00526-007-0124-7


We consider a class of non convex scalar functionals of the form
$$ \mathcal{F}(u) = \int\limits_\Omega f(x,u,Du)\,dx, $$
under standard assumptions of regularity of the solutions of the associated relaxed problem and of local affinity of the bipolar f** of f on the set {f** < f}. We provide an existence theorem, which extends known results to lagrangians depending explicitly on the three variables, by the introduction of integro-extremal minimizers of the relaxed functional which solve the equation
$$ f^{\ast\ast}(x, u,Du) - f(x, u,Du) =0, $$
or the opposite one, almost everywhere and in viscosity sense.

Mathematics Subject Classification

49J45 49L25 

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.S.I.S.S.ATriesteItaly