Calculus of Variations and Geometric Measure Theory

S. Kholmatov - P. Piovano

Existence of minimizers for the SDRI model

created by kholmatov2 on 11 Jun 2020
modified by piovano on 14 Aug 2021


Submitted Paper

Inserted: 11 jun 2020
Last Updated: 14 aug 2021

Year: 2020


The SDRI model introduced in "Sh. Kholmatov, P. Piovano, Arch. Rational Mech. Anal., 238 (2020), 415-488" in the framework of the theory on Stress-Driven Rearrangement Instabilities for the morphology of crystalline material under stress is considered. 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 available results for epitaxially-strained thin films and material cavities. Due to the lack of compactness and lower semicontinuity even for sequences of $m$-minimizers, i.e., energy minimizers among configurations with a fixed number $m$ of connected boundary-components, the minimizing candidate of the SDRI model is directly constructed. By means of uniform density estimates for the local decay of the energy at the $m$-minimizers' boundaries, such candidate is then shown to be a minimizer also in view of the convergence of the energy at $m$-minimizers to the energy infimum as $m\to\infty$. Finally, regularity properties for the morphology satisfied by every minimizer are deduced.

Keywords: existence, regularity, Thin films, SDRI, crystal cavities, minimal configurations, elastic energy , surface energy, fractures, interface instabilities, density estimates