An L Two Well Liouville Theorem

We provide a different approach to and prove a (partial) generalisation of a recent theorem on the structure of low energy solutions of the compatible two well problem in two dimensions [Lor05], [CoSc06]. More specifically we will show that a “quantitative” two well Liouville theorem holds for the set of matrices K = SO (2) ∪ SO (2) H where H = ( σ 0 0 σ−1 ) under a constraint on the L norm of the second derivative. Our theorem is the following. Let p ≥ 1, q > 1. Let u ∈ W 2,p (B1 (0)) ∩ W 1,q (B1 (0)). There exists positive constants C1 << 1,C2 >> 1 depending only on σ, p, q such that if u satisfies the following inequalities ∫ B 1 2 (0) d (Du (z) ,K) dLz ≤ C1ε, ∫ B1(0) ∣D2u (z) ∣∣p dLz ≤ C1ε1−p then there exist A ∈ K such that