Hafedh Ben Belgacem
(2010)

In this paper, we consider a Borel
measurable function on
the space of
${\scriptstyle m\times n}$ matrices ${\scriptstyle f:{M}^{m\times n}\to \overline{\mathbb{R}}}$
taking the value
${\scriptstyle +\infty}$, such that its rank-one-convex
envelope
${\scriptstyle Rf}$ is finite and satisfies for some fixed
${\scriptstyle p>1}$:
$${\scriptstyle -{c}_{0}\le Rf\left(F\right)\le {c(1+\parallel F\parallel}^{p})\phantom{\rule{4pt}{0ex}}\text{for}\phantom{\rule{4.0pt}{0ex}}\text{all}\phantom{\rule{4pt}{0ex}}F\in {M}^{m\times n},}$$
where
${\scriptstyle c,{c}_{0}>0}$. Let ${\scriptstyle \xd8}$ be a given
regular bounded
open domain of
${\scriptstyle {\mathbb{R}}^{n}}$. We define on ${\scriptstyle {W}^{1,p}(\xd8;{\mathbb{R}}^{m})}$
the functional
$${\scriptstyle I\left(u\right)={\int}_{\xd8}f\left(\nabla u\left(x\right)\right)\phantom{\rule{4pt}{0ex}}dx.}$$
Then, under some technical restrictions on
${\scriptstyle f}$, we show that the relaxed functional
${\scriptstyle \overline{I}}$
for the weak topology
of
${\scriptstyle {W}^{1,p}(\xd8;{\mathbb{R}}^{m})}$ has the integral
representation:
$${\scriptstyle \overline{I}\left(u\right)={\int}_{\xd8}Q\left[Rf\right]\left(\nabla u\left(x\right)\right)\phantom{\rule{4pt}{0ex}}dx,}$$
where for a given function ${\scriptstyle g}$,
${\scriptstyle Qg}$ denotes...