Lower semicontinuity for quasiconvex integrals of higher order (Q1294021)

The authors have considered the functional of the type \[ {\mathcal F}(u)= \int_\Omega F(x,u,\dots, D^k u) dx, \] where \(\Omega\) is an open bounded set of \(\mathbb{R}^n\) and \(F\) is a Carathéodory function. By introducing more regular approximating function, the lower-semicontinuity of the above functional with respect to the weak topology \(W^{k,p}(\Omega; \mathbb{R}^m)\) under \(p\)-growth conditions has been proved. It is further shown that for \(k=2\) the quasiconvexity and growth condition imply the local Lipschitz continuity of \(F\).