On the effect of geometry on scaling laws for a class of martensitic phase transformations

We study scaling laws for singular perturbation problems associated with a class of two-dimensional martensitic phase transformations and deduce a domain dependence of the scaling law in the singular perturbation parameter. In these settings the respective scaling laws give rise to a selection principle for specific, highly symmetric domain geometries for the associated nucleation microstructure. More precisely, firstly, we prove a general lower bound estimate illustrating that in settings in which the domain and well geometry are incompatible in the sense of the Hadamard jump condition, then necessarily at least logarithmic losses in the singular perturbation parameter occur in the associated scaling laws. Second, for specific phase transformations in two-dimensional settings we prove that this gives rise to a dichotomy involving logarithmic losses in the scaling law for generic domains and optimal linear scaling laws for very specific, highly compatible polygonal domains. In these situations the scaling law thus gives important insight into optimal isoperimetric domains. We discuss both the geometrically linearized and nonlinear settings.