Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals

AbstractWe study semicontinuity of multiple integrals ∫Ωf(x,u,Du) dx, where the vector-valued function u is defined for $$x \varepsilon \Omega \subset \mathbb{R}^n $$ with values in ℝN. The function f(x,s,ξ) is assumed to be Carathéodory 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 of the Sobolev space H1,p(ΩℝN). 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 beingconvex andindependent of (x,s) for large values of ξ. 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 H1,p(Ωℝn) for small p, in particular for some p smaller than n.