Scaling laws for linear-elastic two-variant martensitic phase transitions in general domains

We consider a two-well problem within geometrically linear elasticity, on a bounded Lipschitz set in dimension $n\ge 2$. We assume that the two wells are compatible through a symmetrized rank-one connection and that the displacement matches their average on the boundary, and we include a regularization based on the total surface energy. We show that there are two distinct regimes, with the energy scaling differently in terms of the surface-energy coefficient $\eps$: it is proportional to $\eps^{2/3}$ if there are two distinct compatible twin-normal directions, and to $\eps^{4/5}$ if there is only one. This matches previous findings in rectangles and other special geometries, mostly in two dimensions, and the lower bounds obtained by a Fourier method by Raiţă, Rüland, Tissot and Tribuzio (2024). Our results are obtained by explicit constructions, which contain a self-similar refinement of oscillations close to the boundary, and by a short real-space proof of the matching lower bounds.