Remarks on the lower semicontinuity of quasiconvex integrals (Q1902878)

The paper provides some results in the direction of the proof of a general lower semicontinuity theorem with respect to weak convergence of Cartesian currents for quasiconvex integrals of the type \({\mathcal F}(u)= \int_\Omega f(Du)dx\) with no growth conditions on the integrand \(f\). Some sequential lower semicontinuity theorems are proved for integrals as above with \(f\) quasiconvex, nonnegative and estimated from above by some powers of the determinants of some submatrices of the matrix of the variables. The topology considered is the one for which \(\{u_h\}\) converges to \(u\) if and only if \(\{u_h\}\) converges to \(u\) in \(L^1(\Omega)\) and the above-mentioned powers of the determinants of certain submatrices of \(\{Du_h\}\) are bounded in \(L^1(\Omega)\). A notion of quasiconvexity in which Cartesian currents are taken into account as test functions is introduced and its equivalence with the sequential weak lower semicontinuity of \(\mathcal F\) in spaces of Cartesian currents is established. Finally, the above described lower semicontinuity results are extended to spaces of Cartesian currents.