Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals. (English) Zbl 0573.49010

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 Carath√©odory 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.
Reviewer: E.Mascolo


49J45 Methods involving semicontinuity and convergence; relaxation
26B25 Convexity of real functions of several variables, generalizations
54C08 Weak and generalized continuity
41A30 Approximation by other special function classes
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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