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

Abstract

We prove the lower semicontinuity of functionals of the form \[ \int \limits_Ω\! V(α) \, \mathrm{d} |\mathrm{E} u| \, , \] with respect to the weak converge of $\alpha$ in $W^{1,\gamma }(\Omega )$, $\gamma >1$, and the weak* convergence of $u$ 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 $\gamma <n$.