A two well Liouville theorem with applications

We will we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller. We then give some applications to the problem of getting a quantitive estimate of how much convex integration solutions oscillate.