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

We establish a general weak* lower semicontinuity result in the space $\text{BD}(\mathrm{\Omega })$ of functions of bounded deformation for functionals of the form $$\text{Fcal}(u):={\int }_{\mathrm{\Omega }}f(x,\text{Ecal}u)\text{dd}x+{\int }_{\mathrm{\Omega }}{f}^{\mathrm{\infty }}(x,\frac{\text{di}{E}^{s}u}{\text{di}\text{abs}{E}^{s}u})\text{dd}\text{abs}{E}^{s}u+{\int }_{\mathrm{\partial }\mathrm{\Omega }}{f}^{\mathrm{\infty }}(x,u{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}\odot n_{\mathrm{\Omega }})\text{dd}{\text{Hcal}}^{d-1}\text{,}$$ $u\in \text{BD}(\mathrm{\Omega })$. 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 $\text{BD}(\mathrm{\Omega })$, which is not available at present. We also include existence and relaxation results for variational problems in $\text{BD}(\mathrm{\Omega })$, as well as a complete discussion of some differential inclusions for the symmetrized gradient in two dimensions.