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We study the functional If (u) = T Ω f(u(x)) dx, where u = (u1, . . . , um) and each uj is constant along some subspace Wj of R . We show that if intersections of the Wj ’s satisfy a certain condition then If 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 {Wj}j=1,...,m to have the equivalence: If is weakly continuous if and only if f is Λ-affine.

On the condition of Λ-convexity in some problems of weak continuity and weak lower semicontinuity