On the formation of microstructure for singularly perturbed problems with two, three, or four preferred gradients (Q6599775)

The author examines a singularly perturbed energy associated with two, three, or four preferred gradients and investigates the formation of microstructures under incompatible Dirichlet boundary conditions. The main result is a scaling law for the minimal energy in the context of these incompatible boundary conditions. Within a unit square, scaling laws are established in relation to two parameters: one quantifying the transition cost between different preferred gradients, and the other assessing the incompatibility between the set of preferred gradients and the boundary conditions. Furthermore, the study demonstrates how simple building blocks and covering arguments can yield upper bounds on the energy, as well as solutions to the differential inclusion problem in general Lipschitz domains.