Relaxation of Functionals in the Space of Vector-Valued Functions of Bounded Hessian

Abstract

In this paper it is shown that if $\Omega \subset \mathbb{R}^N$ is an open, bounded Lipschitz set, and if $f: \Omega \times \mathbb{R}^{d \times N \times N} \rightarrow [0, \infty)$ is a continuous function with $f(x, \cdot)$ of linear growth for all $x \in \Omega$, then the relaxed functional in the space of functions of Bounded Hessian of the energy \[ F[u] = \int_{\Omega} f(x, \nabla^2u(x)) dx \] for bounded sequences in $W^{2,1}$ is given by \[ {\cal F}[u] = \int_\Omega {\cal Q}_2f(x, \nabla^2u) dx + \int_\Omega ({\cal Q}_2f)^{\infty}\bigg(x, \frac{d D_s(\nabla u)}{d |D_s(\nabla u)|} \bigg) d |D_s(\nabla u) |. \] This result is obtained using blow-up techniques and establishes a second order version of the $BV$ relaxation theorems of Ambrosio and Dal Maso and Fonseca and M\"uller. The use of the blow-up method is intended to facilitate future study of integrands which include lower order terms and applications in the field of second order structured deformations.