On the lower semicontinuity of quasiconvex integrals (Q2875934)

Given an integral functional of the form NEWLINE\[NEWLINEF(u)= \int_\Omega f(x, u(x),\nabla u(x))\,dx,NEWLINE\]NEWLINE it is known that weak lower semicontinuity of \(F\) in \(W^{1,p}\) is related to quasiconvexity (in the sense of Morrey) of \(f(x,\xi,\cdot)\). Here, \(u: \Omega\subseteq \mathbb{R}^n\to\mathbb{R}^N\). The present paper introduces a novel and relatively elementary proof of such a result under suitable \(p\)-growth conditions.