A Two Well Liouville Theorem

Abstract

In this paper 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. Let for . Let . Let . Let be a invertible bilipschitz function with , . There exists positive constants and depending only on σ, , such that if and u satisfies the following inequalities then there exists and such that