A lower semicontinuity result for linearised elasto-plasticity coupled with damage in $W^{1,\gamma }$, $\gamma >1$

We prove the lower semicontinuity of functionals of the form $$\underset{\mathrm{\Omega }}{\int }\text{\hspace{0.17em}⁣}V(\alpha )\mathrm{d}\mathrm{\mid }\mathrm{E}u\mathrm{\mid },$$ with respect to the weak converge of $\alpha$ in $W^{1,\gamma }(\mathrm{\Omega })$, $\gamma > 1$,andtheweak\ast convergenceof$u$in$BD(\Omega)$,where$\Omega \subset $\mathbb{R}$^n$. These functional arise in the variational modelling of linearised elasto-plasticity coupled with damage and their lower semicontinuity is crucial in the proof of existence of quasi-static evolutions. This is the first result achieved for subcritical exponents $\gamma <n$.