Calculus of Variations and Geometric Measure Theory

E. Stepanov

Relaxation and convexity of functionals with pointwise nonlocality

created on 25 Nov 2001
modified on 15 Feb 2003


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


It is shown that the relaxation of the integral functional involving argument deviations $$ I(u):=\int\Omega f(x,\{ui(g{ij}(x))\}{i,j=1}{k,l})\, d\mu\Omega(x), $$ 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