Approximation of Quasiconvex Functions, and Lower Semicontinuity of Multiple Integrals

We study semicontinuity of multiple integrals ~ f~,u,Du)dx~ where the vector-valued function u is defined for xE ~C N with values in ~ . The function f(x,s,~) is assumed to be Carath@odory and quasiconvex in Morrey's sense. We give conditions on the growth of f that guarantee the sequential lower semicontinuity of the given integral in the weak topology 1 N of the Sobolev space H 'P(~;~ ). The proofs are based on some approximation results for f. In particular we can approximate f by a nondecreasing sequence of quasiconvex functions, each of them being convex and independent of (x,s) for large values of ~. In the special polyconvex case, for example if n: N and f(Du) is equal to a convex function of the Jacobian detDu, then we obtain semicontinuity in the weak topology of 1,p(~ n H ;~ ) for small p, in particular for some p smaller than n.