Relaxation and convexity of functionals with pointwise nonlocality

It is shown that the relaxation of the integral functional involving argument deviations I(u):=∫f(x{ui(gij))}k,li,j=1)d μΩ(x), in weak topology of a Lebesgue space (Lp(⊖, μμ))k (where (Ω, ∑(Ω), μΩ) and (⊖, ∑(⊖),μ⊖) are standard measure spaces, the latter with nonatomic measure), coincides with its convexification whenever the matrix of measurable functions gij: Ω → ⊖ satisfies the special condition, called unifiability, which can be regarded as collective nonergodicity or commensurability property, and is automatically satisfied only if k = l = 1. If, however, either k > 1 or l > 1, then it is shown that as opposed to the classical case without argument deviations, for nonunifiable function matrix {gij} one can always construct an integrand f so that the functional I itself is already weakly lower semicontinuous but not convex.