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

TL;DR

This paper characterizes the relaxation of second-order energy functionals in the space of functions with bounded Hessian, extending classical BV relaxation results to second derivatives using blow-up techniques.

Contribution

It provides a second order relaxation theorem for functionals involving the Hessian, generalizing BV relaxation results to the space of functions with bounded Hessian.

Findings

Explicit formula for the relaxed functional involving quasiconvexification.

Extension of BV relaxation theorems to second order derivatives.

Application framework for second order structured deformations.

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.