Center for                           Nonlinear Analysis CNA Home People Seminars Publications Workshops and Conferences CNA Working Groups CNA Comments Form Summer Schools Summer Undergraduate Institute PIRE Cooperation Graduate Topics Courses SIAM Chapter Seminar Positions Contact Publication 18-CNA-002 Relaxation of Functionals in the Space of Vector-Valued Functions of Bounded Hessian Adrian HagertyDepartment of Mathematical Sciences Carnegie Mellon University Pittsburgh, PAahagerty@andrew.cmu.eduAbstract: 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