On Lambda-Convexity Conditions in the Theory of Lower Semicontinuous Functionals

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.