On the lower semicontinuity of supremal functional under differential constraints

Abstract

We study the weak* lower semicontinuity of functionals of the formThe notion of A-Young quasiconvexity, which is introduced here, provides a sufficient condition when f (x, •) is only lower semicontinuous. We also establish necessary conditions for weak* lower semicontinuity. Finally, we discuss the divergence and curl-free cases and, as an application, we characterise the strength set in the context of electrical resistivity.

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

“…x ∈ Ω and for every Σ ∈ M d×N . This problem has been studied by Garroni-Nesi-Ponsiglione in the special case f (x, Σ) = λ(x)|Σ| (see [17]), by Champion-De Pascale-Prinari when f satisfies a generalized Jensen inequality for gradient Young measures (see [12]) and by Bocea-Nesi and Ansini-Prinari when V ∈ L ∞ (Ω; M d×N ) is constrained to satisfy a more general rank-constant differential constraint (see [1,10]). In this paper we give a complete representation of the Γ-limit by showing that, as p → ∞, the family (F p ) p Γ-converges, with respect to the uniform convergence, to the supremal functional F…”

“…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 the following theorem we derive the rank-one level convexity as a necessary condition for the weak* lower semicontinuity of a supremal functional. A version of this theorem has been given in [2,Theorem 6.2], in order to characterize the weak* lower semicontinuity of functionals of the form…”

“…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).…”

Power-Law Approximation under Differential Constraints

“…Moreover, in the Example 5.9 we show that such inclusion is strict. In a forthcoming paper [4] we perform a deeper analysis in order to find necessary and sufficient conditions for the lower semicontinuity of supremal functional of the form (1.7).…”