Relaxation and convexity of functionals with pointwise nonlocality

It is shown that the relaxation of the integral functional involving argument deviations

in weak topology of a Lebesgue space $(L^p(\Theta,\mu_\Theta))^k$ (where $(\Omega,\Sigma(\Omega),\mu_\Omega)$ and $(\Theta,\Sigma(\Theta),\mu_\Theta)$are standard measure spaces, the latter with nonatomic measure), coincides with its convexification whenever the matrix of measurable functions $g_{ij}$: $\Omega\to \Theta$ 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 $\{g_{ij}\}$ one can always construct an integrand $f$ so that the functional $I$ itself is already weakly lower semicontinuous but not convex.