On the definition and the lower semicontinuity of certain quasiconvex integrals (Q1086828)

The author studies the lower semicontinuity in the weak topology of \(H^{1,P}(\Omega;R^ N)\) of the functional \[ I(u)=\int_{\Omega}f(x,Du(x))dx, \] where f(x,z) is a continuous function in \(\Omega \times R^{nN}\), quasiconvex with respect on z, that satisfies for some \(p\leq q\) the following growth condition \(c_ 1| z|^ p\leq f(x,z)\leq c_ 2(1+| z|)^ q)\). Since I(u) is well defined if \(u\in H^{1,q}(\Omega;R^ N)\), the author has first to extend the integral to functions \(u\in H^{1,P}(\Omega,R^ N)\) and after that he studies the semicontinuity in order to obtain existence of minima.