## 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 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.
Reviewer: E.Mascolo

### MSC:

 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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### References:

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