A variational model allowing both smooth and sharp phase boundaries in solids

We present models for solid-solid phase transitions with surface energy that allow both smooth and sharp interfaces. The models involve the minimisation of an energy that consists of three terms: the elastic energy (a double-well potential), the smooth-interface surface energy and the sharp-interface surface energy. Existence of solutions is shown in arbitrary dimensions. The second part of the paper deals with the one-dimensional case. For the first 1D model (in which the sharp-interface energy is the same regardless of the size of the jump of the gradient), we study the regime of the parameters (one parameter represents the boundary conditions, one models the energy of the sharp interface, and the third one models the energy of the smooth interfaces) for which the minimiser presents smooth interfaces, sharp interfaces or no interfaces. We also prove that a suitable scaling of the functional $\Gamma$-converges to a pure sharp-interface model, as the parameters penalising the formation of interfaces go to zero. For the second 1D model (in which the sharp-interface energy depends on the size of the jump and can tend to zero as the jump tends to zero), we describe general properties of the minimisers, and show that their gradients have a finite number of discontinuity points.