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The model introduced in [Kholmatov-Piovano 2020] in the framework of thetheory on Stress-Driven Rearrangement Instabilities (SDRI) [Asaro-Tiller 1972;Grinfeld 1993} for the morphology of crystalline materials under stress isconsidered. As in [Kholmatov-Piovano 2020] and in agreement with the models in[Lowengrub et al. 2009; Spencer 1999], a mismatch strain, rather than aDirichlet condition as in [Crismale-Friedrich 2020], is considered to includeinto the analysis the lattice mismatch between the crystal and possibleadjacent (supporting) materials. The existence of solutions is established indimension two in the absence of graph-like assumptions and of the restrictionto a finite number $m$ of connected components for the free boundary of theregion occupied by the crystalline material, thus extending previous resultsfor epitaxially strained thin films and material cavities. Due to the lack ofcompactness 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 aminimizer 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 areestablished.

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