The energy scaling behavior of a class of incompatible two-well problems

In this article, we study scaling laws for singularly perturbed two-well energies with prescribed Dirichlet boundary data in settings where the wells and/or the boundary data are incompatible. Our main focus is the geometrically linear two-well problem, for which we characterize the energy scaling in two dimensions for nearly all combinations of linear boundary data and stress-free strains. In particular, we prove that if the boundary data enforces oscillations and the weight ε of the surface energy is small, the minimal energy upon subtracting the zeroth-order contribution scales either as ε^{4/5} or as ε^{2/3}, depending on whether the wells differ by a rank-one or a rank-two matrix, respectively. For the gradient and divergence-free two-well problem, we obtain analogous results, showing an ε^{2/3}-scaling behavior in two dimensions whenever oscillations are energetically favored. These results follow by deriving matching upper and lower scaling bounds. The lower scaling bounds are established in a general A-free framework for incompatible two-well problems, which allows us to compute the excess energy and characterize boundary data which enforce oscillations. The upper scaling bounds are obtained by branching constructions which are adapted to the incompatible setting.