On the condition of ${\mit \Lambda }$-convexity in some problems of weak continuity and weak lower semicontinuity
Volume 89 / 2001
Colloquium Mathematicum 89 (2001), 43-59
MSC: 49J45, 35E10.
DOI: 10.4064/cm89-1-3
Abstract
We study the functional $I_f(u)=\int_{{\mit\Omega}} f(u(x))\,dx$, where $u=(u_1, \ldots ,u_m)$ and each $u_j$ is constant along some subspace $W_j$ of ${\mathbb R}^{n} $. We show that if intersections of the $W_j$'s satisfy a certain condition then $I_f$ is weakly lower semicontinuous if and only if $f$ is ${\mit\Lambda} $-convex (see Definition 1.1 and Theorem 1.1). We also give a necessary and sufficient condition on $\{ W_j\}_{j=1, \ldots ,m}$ to have the equivalence: $I_f$ is weakly continuous if and only if $f$ is ${\mit\Lambda} $-affine.
