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

TL;DR

This paper extends mathematical results on the existence and approximation of minimizers for energy models of epitaxially strained films and voids within three-dimensional linear elasticity, introducing a new convergence concept for complex sets.

Contribution

It generalizes previous work by removing integrability assumptions and introduces a novel convergence method for jump sets of GSBD^p functions in elasticity models.

Findings

Established existence of minimizers in 3D linear elasticity models.

Developed a new convergence notion for rectifiable sets in elasticity.

Extended relaxation and approximation results for energy functionals.

Abstract

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.