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On the definition and the lower semicontinuity of certain quasiconvex integrals. (English) Zbl 0609.49009
The author studies the lower semicontinuity in the weak topology of $$H^{1,P}(\Omega;R^ N)$$ of the functional $I(u)=\int_{\Omega}f(x,Du(x))dx,$ where f(x,z) is a continuous function in $$\Omega \times R^{nN}$$, quasiconvex with respect on z, that satisfies for some $$p\leq q$$ the following growth condition $$c_ 1| z|^ p\leq f(x,z)\leq c_ 2(1+| z|)^ q)$$. Since I(u) is well defined if $$u\in H^{1,q}(\Omega;R^ N)$$, the author has first to extend the integral to functions $$u\in H^{1,P}(\Omega,R^ N)$$ and after that he studies the semicontinuity in order to obtain existence of minima.
Reviewer: R.Schianchi

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 26B25 Convexity of real functions of several variables, generalizations 35J50 Variational methods for elliptic systems
##### Keywords:
quasiconvexity; lower semicontinuity; existence of minima
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##### References:
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