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

In this paper it is shown that if $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is an open, bounded Lipschitz set, and if $f: \Omega \times $\mathbb{R}$^{d \times N \times N} \rightarrow [0, \infty)$isacontinuousfunctionwith$f(x, \cdot)$ of linear growth for all $x\in \mathrm{\Omega }$, then the relaxed functional in the space of functions of Bounded Hessian of the energy $$F[u]={\int }_{\mathrm{\Omega }}f(x,{\mathrm{\nabla }}^{2}u(x))dx$$ for bounded sequences in $W^{2,1}$ is given by $$\mathcal{F}[u]={\int }_{\mathrm{\Omega }}{\mathcal{Q}}_{2}f(x,{\mathrm{\nabla }}^{2}u)dx+{\int }_{\mathrm{\Omega }}({\mathcal{Q}}_{2}f{)}^{\mathrm{\infty }}(x,\frac{d{D}_s(\mathrm{\nabla }u)}{d\mathrm{\mid }{D}_s(\mathrm{\nabla }u)\mathrm{\mid }})d\mathrm{\mid }{D}_s(\mathrm{\nabla }u)\mathrm{\mid }\mathrm{.}$$ 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.