On the relaxation of nonconvex superficial integral functionals
Résumé
We present a new approach to the variational relaxation of functionals $F:D(\mathbb{R}^N;\mathbb{R}^m)\to[0,\infty[$ of the type:
$$
F(v):=\int_{\mathbb{R}^N}W(\nabla v(x))d\mu(x),
$$
where $W:\mathbb{R}^{mN}\to[0,\infty[$ is a continuous function with growth conditions of order $p\geq 1$ but not necessarily convex. We essentially study the case when $\mu$ is the $k$-dimensional Hausdorff measure restricted to a suitable piece of a $k$-dimensional smooth submanifold of $\mathbb{R}^N$.