We study the functional $I_f\left(u\right)={\int }_{\Omega }f\left(u\left(x\right)\right)dx$, where u=(u₁, ..., uₘ) and each $u_j$ is constant along some subspace $W_j$ of ℝⁿ. 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 Λ-convex (see Definition 1.1 and Theorem 1.1). We also give a necessary and sufficient condition on ${W_j}_{j=1,...,m}$ to have the equivalence: $I_f$ is weakly continuous if and only if f is Λ-affine.

This article surveys results on the global surjectivity of linear partial differential operators with constant coefficients on the space of real analytic functions. Some new results are also included.