Relaxation and convexity of functionals with pointwise nonlocality

Abstract:

It is shown that the relaxation of the integral functional involving argument deviations \[ I(u):=\int _\Omega f(x,\{u_i(g_{ij}(x))\}_{i,j=1}^{k,l}) d\mu _\Omega (x), \] 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.