Equilibrium configurations for epitaxially strained films and material voids in three-dimensional linear elasticity

We extend the results about existence of minimizers, relaxation, and approximation proven by Chambolle et al. in 2002 and 2007 for an energy related to epitaxially strained crystalline films, and by Braides, Chambolle, and Solci in 2007 for a class of energies defined on pairs of function-set. We study these models in the framework of three-dimensional linear elasticity, where a major obstacle to overcome is the lack of any 'a priori' assumption on the integrability properties of displacements. As a key tool for the proofs, we introduce a new notion of convergence for $(d{-}1)$-rectifiable sets that are jumps of $GSBD^p$ functions, called $\sigma^p_{\mathrm{sym}}$-convergence.