Publication 18-CNA-002
Relaxation of Functionals in the Space of Vector-Valued Functions of Bounded Hessian
Adrian Hagerty
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA
ahagerty@andrew.cmu.edu
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.
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