Lower semicontinuity for integral functionals in the space of functions of bounded deformation via rigidity and Young measures

Abstract

We establish a general weak* lower semicontinuity result in the space $\BD(\Omega)$ of functions of bounded deformation for functionals of the form $$\Fcal(u) := \int_\Omega f \bigl(x, \Ecal u \bigr) \dd x + \int_\Omega f^\infty \Bigl(x, \frac{\di E^s u}{\di \abs{E^s u}} \Bigr) \dd \abs{E^s u} + \int_{\partial \Omega} f^\infty \bigl(x, u|_{\partial \Omega} \odot n_\Omega \bigr) \dd \Hcal^{d-1}$$, $u \in \BD(\Omega)$. The main novelty is that we allow for non-vanishing Cantor-parts in the symmetrized derivative $Eu$. The proof is accomplished via Jensen-type inequalities for generalized Young measures and a construction of good blow-ups, which is based on local rigidity arguments for some differential inclusions involving symmetrized gradients. This strategy allows us to establish the lower semicontinuity result without an Alberti-type theorem in $\BD(\Omega)$, which is not available at present. We also include existence and relaxation results for variational problems in $\BD(\Omega)$, as well as a complete discussion of some differential inclusions for the symmetrized gradient in two dimensions.