Existence of Minimizers for NonLevel Convex Supremal Functionals

Abstract

The paper is devoted to determine necessary and sufficient conditions for existence of solutions to the problem inf ess sup(Ω) , when the supremand f is not necessarily level convex. These conditions are obtained through a comparison with the related level convex problem and are written in terms of a differential inclusion involving the boundary datum. Several conditions of convexity for the supremand f are also investigated. RésuméDans cet article onétudie des conditions nécessaires et suffisantes pour l'existence de solutions pour le problème de minimisation inf ess sup x∈Ω f (∇u(x)) : u ∈ u0 + W 1,∞ 0 (Ω) lorsque la fonction f n'est pas une fonction convexe par niveaux. La stratégie utilisée pour obtenir ces conditions est celle de comparer ce problème avec son problème relaxé. On obtient comme condition nécessaire et suffisante une inclusion différentielle sur la donnée au bord. Onétudie aussi plusieurs conditions de convexité.

On the lower semicontinuity and approximation of $${L^{\infty}}$$ L ∞ -functionals

“…Any upper semicontinuous and weak Morrey quasi-convex function is rank-one level convex (see [1,Proposition 5.2]) but in general the only weak Morrey quasiconvexity is not sufficient to provide the rank-one quasiconvexity: in fact in [24,Remark 5.2], the authors exhibit an example of a lower semicontinuous and weak Morrey quasi-convex function which is not rank-one level convex. In the following theorem we derive the rank-one level convexity as a necessary condition for the weak* lower semicontinuity of a supremal functional.…”

“…In order to apply the direct method of the calculus of variations the main issue is the lower semicontinuity of F . Semicontinuity properties for supremal functionals have recently been studied by many authors; we refer for instance to Barron-Jensen [6], Barron-Jensen-Wang [8], Prinari [22,23] and to the recent papers by Ansini-Prinari [2] and Ribeiro-Zappale [24]. In the context of supremal functionals, in [8] Barron, Jensen and Wang introduce the notion of the weak Morrey quasiconvexity as the natural extension of the notion of Morrey quasiconvexity (see [14] and references therein).…”

“…Moreover, in the scalar case, i.e. d = 1 or N = 1, this condition is also sufficient since it coincides with the notion of level convexity (for a proof, see Theorem 5.5 (ii) and (iii) in [24]). We recall that f : R k → R is level convex if for every t ∈ R the level set ξ ∈ R k : f (ξ) ≤ t is convex.…”

A minimisation problem in L∞with PDE and unilateral constraints

“…Interesting results regarding L ∞ variational problems can be found e.g. in [10,14,15,16,17,18,19,28,43,46,47,48].…”

Inverse Optical Tomography through PDE Constrained Optimization $L^\infty$

“…Interesting theory and applications of L ∞ variational problems can be found e.g. in [10,14,15,16,17,18,19,28,45,47,48,49].…”