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

Author:

Kholmatov Shokhrukh Y.1ORCID,Piovano Paolo2ORCID

Affiliation:

1. Faculty of Mathematics , University of Vienna , Oskar-Morgenstern Platz 1, 1090 Vienna , Austria

2. Dipartimento di Matematica , Politecnico di Milano , Piazza Leonardo da Vinci 32, 20133 Milano , Italy (MUR Excellence Department 2023–2027) ; and WPI c/o Research Platform MMM “Mathematics-Magnetism-Materials”, Fakultät für Mathematik – Universität Wien, 1090 Vienna

Abstract

AbstractThe model introduced in [45] in the framework of the theory on stress-driven rearrangement instabilities (SDRI) [3, 43] for the morphology of crystalline materials under stress is considered. As in [45] and in agreement with the models in [50, 55], a mismatch strain, rather than a Dirichlet condition as in [19], is included into the analysis to represent 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 numbermof 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 [6, 35, 34, 45]. Due to the lack of compactness and lower semicontinuity for the sequences ofm-minimizers, i.e., minimizers among configurations with at mostmconnected boundary components, a minimizing candidate is directly constructed, and then shown to be a minimizer by means of uniform density estimates and the convergence ofm-minimizers’ energies to the energy infimum asm{m\to\infty}. Finally, regularity properties for the morphology satisfied by every minimizer are established.

Funder

Austrian Science Fund

Vienna Science and Technology Fund

Bundesministerium für Bildung, Wissenschaft und Forschung

Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,Analysis

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