On the lower semicontinuity of supremal functional under differential constraints

Ansini, Nadia; Prinari, Francesca

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

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

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

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

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

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