*Published Paper*

**Inserted:** 25 nov 2001

**Last Updated:** 15 feb 2003

**Journal:** Proc. Amer. Math. Soc.

**Volume:** 130

**Number:** 2

**Pages:** 433-442

**Year:** 2002

**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 already the functional $I$ itself be weakly
lower semicontinuous but not convex.

**Keywords:**
relaxation, Nonlocal functionals, composition operator

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