Calculus of Variations and Geometric Measure Theory

N. Ansini - F. Prinari

On the lower semicontinuity of supremal functional under differential constraints

created by ansini on 04 Mar 2014
modified on 15 May 2016


Published Paper

Inserted: 4 mar 2014
Last Updated: 15 may 2016

Journal: ESAIM Control Optim. Calc. Var.
Volume: 21
Number: 4
Pages: 1053-1075
Year: 2015


We study the weak$^*$ lower semicontinuity of functionals of the form \[ F(V)=\mbox{ess sup}_{x \in \Omega} f(x,V (x)) \] where $\Omega\subset \mathbb{R}^N$ is a bounded open set, $V\in L^{\infty}(\Omega;\mathbb{M}^{d\times N})\cap \hbox {Ker} \cal A$ and $\cal A$ is a constant-rank partial differential operator. The notion of $\cal A$-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.