Relaxation of singular functionals defined on Sobolev spaces

Abstract

In this paper, we consider a Borel measurable function on the space of $\scriptstyle m\times n$ matrices $\scriptstyle f: M^{m\times n}\rightarrow \bar{\mathbb{R}}$ taking the value $ \scriptstyle +\infty$, such that its rank-one-convex envelope $\scriptstyle Rf$ is finite and satisfies for some fixed $\scriptstyle p>1$: $$\scriptstyle -c_0\leq Rf(F)\leq c(1+\Vert F\Vert^p)\ \hbox{for all}\ F\in M^{m\times n},$$ where $\scriptstyle c,c_0>0$. Let $\scriptstyle\O$ be a given regular bounded open domain of $\scriptstyle \mathbb{R}^n$. We define on $\scriptstyle W^{1,p}(\O;\mathbb{R}^m)$ the functional $$\scriptstyle I(u)=\int_{\O}f(\nabla u(x))\ dx.$$ Then, under some technical restrictions on $\scriptstyle f$, we show that the relaxed functional $\scriptstyle\bar I$ for the weak topology of $\scriptstyle W^{1,p}(\O;\mathbb{R}^m)$ has the integral representation: $$\scriptstyle \bar I(u)=\int_{\O}Q[Rf](\nabla u(x))\ dx,$$ where for a given function $\scriptstyle g$, $\scriptstyle Qg$ denotes its quasiconvex envelope.