An Existence Result for a Class of Non Convex Problems of the Calculus of Variations
Giulia Treu · Journal of convex analysis · 1998
We consider the functional ∫ Ω [h(γK(∇u(x))) + u(x)] dx u(x) ∈ W¹,¹₀(Ω) where γK 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 ∫ Ω [f(∇u(x)) + u(x)] dx u(x) ∈ W¹,¹₀(Ω).