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

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 ( \(N=2\) $N=2$ ), a continuum approximation for the \(J_1-J_3\) $J_1-J_3$ -model for discrete spin systems ( \(N=4\) $N=4$ ), 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.