
Journal of Convex Analysis 10 (2003), No. 2, 419436 Copyright Heldermann Verlag 2003 On LambdaConvexity Conditions in the Theory of Lower Semicontinuous Functionals Agnieszka Kalamajska Institute of Mathematics, Warsaw University, ul. Banacha 2, 02097 Warszawa, Poland, kalamajs@mimuw.edu.pl [Abstractpdf] 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 rankone conjecture of Morrey. Keywords: Lower semicontinuity, quasiconvexity, Young measures. MSC 2000: 49J45, 49J10, 35E10. FullTextpdf (538 K) 