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

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