Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals (Q1063225)

The author studies the semicontinuity of the following functional in the calculus of variations: \(F(u)=\int_{G}f(x,u,Du)dx\) where u is a vector- valued function. The function f(x,s,\(\xi)\) is assumed to verify the Carathéodory property and to be quasi-convex in Morrey's sense. Growth conditions for f are given to ensure that F is sequentially lower semicontinuous in the weak topology of \(H^{1,p}(G,R^ N)\). The proof is based on some interesting approximation results for f. In particular, it is possible to approximate f with a non decreasing sequence of quasi- convex functions \(f_ K\), which are convex and independent of (x,s) for \(| \xi | \geq K\). Finally, polyconvex functionals in Ball's sense are considered and some counterexamples are given.