Journal of Convex Analysis 10 (2003), No. 2, 419--436
Copyright Heldermann Verlag 2003
Agnieszka Kalamajska
Institute of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warszawa, Poland, kalamajs@mimuw.edu.pl
Consider the functional $I_f(u)=\int_\Omega f(u(x))\, dx$, where $% u=(u_1,\dots ,u_m)$. Assume additionally that each $u_j$ is constant along $W_j$, some subspace of ${\bf R}^n$. We find the family of cones $\Lambda$ in ${\bf R}^m$ such that every $\Lambda$-convex function $f$ defines a functional $I_f$ which is lower semicontinuous under the sequential weak $*$ convergence in $L^\infty (\Omega ,{\bf R}^m )$. Then we apply our result to functionals acting on distributional kernels of differential operators. We also discuss the relations of our problem to the rank--one conjecture of Morrey.
Keywords: Lower semicontinuity, quasiconvexity, Young measures.
MSC 2000: 49J45, 49J10, 35E10.
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