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Publication 18-CNA-002
Adrian Hagerty 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 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.Get the paper in its entirety as 18-CNA-002.pdf |