*Published Paper*

**Inserted:** 5 feb 2002

**Last Updated:** 17 jul 2002

**Journal:** J. Convex Anal.

**Volume:** 8

**Number:** 2

**Pages:** 447-469

**Year:** 2001

**Abstract:**

We are studying the relaxation of
the integral functional involving argument deviations
$$
I(u):=\int_{\Omega} f(x,u(g_{1} (x)),\ldots, u(g_{k} (x)))\, dx,
$$
in weak topology of a Lebesgue space $L^p(\Omega)$, $1<p<+\infty$,
with open bounded $\Omega\subset\R^n$.
It is proven that, unlike the classical case without deviations,
the relaxed functional in general cannot be obtained as convexification
of the original one. However, we show that if the set functions
$g_i$: $\Omega\to \Omega$ satisfies certain condition
(called *unifiability*), which is just a natural extension
of nonergodicity property of a single function
to sets of functions, and which is
automatically satisfied when $k=1$,
then the relaxed functional is equal to the
convexification of the original one. We show that the
unifiability requirement is essential for such a convexification
result for a generic integrand. Further slightly restricting
this condition, we also obtain the nice
representation of the relaxed functional in terms of
convexification of some new integrand, but involving in general
countably many new argument deviations.

**Keywords:**
relaxation, nonlocal functional, shift operator, ergodicity

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