Energy barriers for boundary nucleation in a two-well model without gauge invariance

We study energy scaling laws for a simplified, singularly perturbed, double-well nucleation problem confined in a half-space, in the absence of gauge invariance and for an inclusion of fixed volume. Motivated by models for boundary nucleation of a single-phase martensite inside a parental phase of austenite, our main focus in this nonlocal isoperimetric problem is how the relationship between the rank-1 direction and the orientation of the half-space influences the energy scaling with respect to the fixed volume of the inclusion. Up to prefactors depending on this relative orientation, the scaling laws coincide with the corresponding ones for bulk nucleation \cite{knupfer2011minimal} for all rank-1 directions, \textit{but} the ones normal to the confining hyperplane, where the scaling is as in a three-well problem in full space, resulting in a lower energy barrier \cite{Tribuzio-Rueland_1}.