On the lower semicontinuity of supremal functional under differential constraints

Ansini, Nadia; Prinari, Francesca

doi:10.1051/cocv/2014058

Ansini, Nadia ^1,² ; Prinari, Francesca ³

¹ Dip. di Matematica, Sapienza Università di Roma, P.le Aldo Moro 2, 00185 Rome, Italy.
² Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK
³ Dip. di Matematica e Informatica, Università di Ferrara, Via Machiavelli 35, 44121 Ferrara, Italy.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 4, pp. 1053-1075.

Résumé

We study the weak* lower semicontinuity of functionals of the form

F (V) =_{x \in} f (x, V (x)),

where $Ω \subset ℝ^{N}$ is a bounded open set, $V \in L^{\infty} (Ω;) \cap Ker$ and �� is a constant-rank partial differential operator. The notion of ��-Young quasiconvexity, which is introduced here, provides a sufficient condition when $f (x, \cdot)$ 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.

Reçu le : 2014-05-05

MR Zbl

DOI : 10.1051/cocv/2014058

Classification : 49J45, 35E99
Mots-clés : Supremal functionals, Γ-convergence, L^p-approximation, lower semicontinuity, 𝒜-quasiconvexity

Affiliations des auteurs :

Ansini, Nadia ^{1, 2} ; Prinari, Francesca ³

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Ansini, Nadia; Prinari, Francesca. On the lower semicontinuity of supremal functional under differential constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 4, pp. 1053-1075. doi : 10.1051/cocv/2014058. http://archive.numdam.org/articles/10.1051/cocv/2014058/

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