Calculus of Variations and Geometric Measure Theory

E. Stepanov - A. Zdorovtsev

Relaxation of some nonlocal integral functionals in weak topology of Lebesgue spaces

created on 05 Feb 2002
modified on 17 Jul 2002


Published Paper

Inserted: 5 feb 2002
Last Updated: 17 jul 2002

Journal: J. Convex Anal.
Volume: 8
Number: 2
Pages: 447-469
Year: 2001


We are studying the relaxation of the integral functional involving argument deviations $$ I(u):=\int\Omega f(x,u(g1 (x)),\ldots, u(gk (x)))\, dx, $$ 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