On the lower semicontinuity of supremal functional under differential constraints

¹ Dip. di Matematica, Sapienza Università di Roma, P.le Aldo Moro 2, 00185 Rome, Italy.
ansini@mat.uniroma1.it
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.
prnfnc1@unife.it

Received: 5 May 2014

Abstract

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

where Ω ⊂ ℝ^N is a bounded open set, and 𝒜 is a constant-rank partial differential operator. The 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.