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Young measure approach to characterization of behaviour of integral functionals on weakly convergent sequences by means of their integrands. (English) Zbl 0923.49009
The integral functional \(I(u)= \int_\Omega L(x,u(x),\nabla u(x))dx\) with \(u\in W^{1,p} (\Omega; \mathbb{R}^m)\), admitting also \(I(u)= +\infty\), is addressed. The sequential weak lower semicontinuity of \(I\) is related with quasiconvexity of \(L(x,u,\cdot)\). Moreover, the property that \(\{u_k\}\) converges weakly to \(u\) and \(I(u_k)\to I(u)\) implies \(\{u_k\}\) converging strongly is related with a so-called strict \(p\)-quasiconvexity. Young measure theory, presented in some detail in this paper, and equi-integrability are main tools used also to give alternative proofs of some other already known results.

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
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