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 $f:\Omega \times {\mathbb{R}}^{d\times N\times N}\to [0,\infty )$ 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 \[ F[u] = \int_Ω f(x, \nabla^2u(x)) dx \] for bounded sequences in $W^{2,1}$ is given by \[ {\cal F}[u] = \int_Ω{\cal Q}_2f(x, \nabla^2u) dx + \int_Ω({\cal Q}_2f)^{\infty}\bigg(x, \frac{d D_s(\nabla u)}{d |D_s(\nabla u)|} \bigg) d |D_s(\nabla u) |. \] 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 Müller. 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.

49 pages