Existence of minimizers for the SDRI model in 2d: wetting and dewetting regime with mismatch strain

The model introduced in [Kholmatov-Piovano 2020] in the framework of the theory on Stress-Driven Rearrangement Instabilities (SDRI) [Asaro-Tiller 1972; Grinfeld 1993} for the morphology of crystalline materials under stress is considered. As in [Kholmatov-Piovano 2020] and in agreement with the models in [Lowengrub et al. 2009; Spencer 1999], a mismatch strain, rather than a Dirichlet condition as in [Crismale-Friedrich 2020], is considered to include into the analysis the lattice mismatch between the crystal and possible adjacent (supporting) materials. The existence of solutions is established in dimension two in the absence of graph-like assumptions and of the restriction to a finite number $m$ of connected components for the free boundary of the region occupied by the crystalline material, thus extending previous results for epitaxially strained thin films and material cavities. Due to the lack of compactness and lower semicontinuity for the sequences of $m$-minimizers, i.e., minimizers among configurations with at most $m$ connected boundary components, a minimizing candidate is directly constructed, and then shown to be a minimizer by means of uniform density estimates and the convergence of $m$-minimizers' energies to the energy infimum as $m\to\infty$. Finally, regularity properties for the morphology satisfied by every minimizer are established.