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

Article

DOI: 10.1007/s00526-007-0124-7

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

## Abstract

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.

49J45 49L25
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