Abstract

We consider the functional ∫Ω[h(γK(∇u(x)))+u(x)] dxu(x)∈W01,1(Ω)\int_\Omega [h(\gamma_K(\nabla u(x))) + u(x)]\,dx \quad u(x) \in W^{1,1}_0(\Omega) where γK\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 ∫Ω[f(∇u(x))+u(x)] dxu(x)∈W01,1(Ω)\int_\Omega [f(\nabla u(x)) + u(x)]\,dx \quad u(x) \in W^{1,1}_0(\Omega).

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Giulia Treu

Dipartimento di Matematica e Informatica, Università di Udine, via delle Scienze 206, 33100 Udine, Italy

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Suggested citation

G. Treu. “An Existence Result for a Class of Non Convex Problems of the Calculus of Variations.” Journal of Convex Analysis 5 (1998), No. 1, 31–44. https://doi.org/10.68381/jca05-3

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Published by Heldermann Verlag, 1998. Rights now held by Banach Press LLC.