On the Formation of Microstructure for Singularly Perturbed Problems with Two, Three, or Four Preferred Gradients

• Published in: Journal of nonlinear science Vol. 34; no. 5; p. 90
• Main Author: Ginster, Janusz
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2024; Springer Nature B.V
• Subjects: Analysis; Approximation; Boundary conditions; Classical Mechanics; Economic Theory/Quantitative Economics/Mathematical Methods; Energy; Flat surfaces; Incompatibility; Martensitic transformations; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Microstructure; Scaling laws; Shape memory alloys; Singular perturbation; Theoretical; Transition costs; Upper bounds; Scaling law; Singular perturbation; Formation of microstructure; 49J40; preferred gradients; 74N15; 82B21
• ISSN: 0938-8974; 1432-1467; 1432-1467
• DOI: 10.1007/s00332-024-10067-x

In this manuscript, singularly perturbed energies with 2, 3, or 4 preferred gradients subject to incompatible Dirichlet boundary conditions are studied. This extends results on models for martensitic microstructures in shape memory alloys ( N = 2 ), a continuum approximation for the J 1 - J 3 -model for discrete spin systems ( N = 4 ), and models for crystalline surfaces with N different facets (general N ). On a unit square, scaling laws are proved with respect to two parameters, one measuring the transition cost between different preferred gradients and the other measuring the incompatibility of the set of preferred gradients and the boundary conditions. By a change of coordinates, the latter can also be understood as an incompatibility of a variable domain with a fixed set of preferred gradients. Moreover, it is shown how simple building blocks and covering arguments lead to upper bounds on the energy and solutions to the differential inclusion problem on general Lipschitz domains.