An Existence Result for a Class of Non Convex Problems of the Calculus of Variations

We consider the functional $\int_\Omega [h(\gamma_K(\nabla u(x))) + u(x)]\,dx \quad u(x) \in W^{1,1}_0(\Omega)$ where $\gamma_K$ is the gauge function of a convex set K and h : [0, ∞[ → [0, ∞] is a possibly non convex function. In the case K ⊂ ℝ² is a closed polytope and Ω ⊂ ℝ² is a bounded convex set we provide a sufficient condition for the existence of the minimum. Besides, as a corollary, we give conditions on Ω ⊂ ℝ² and f : ℝ² → [0, ∞] that are sufficient to the existence of a minimizer of $\int_\Omega [f(\nabla u(x)) + u(x)]\,dx \quad u(x) \in W^{1,1}_0(\Omega)$.