MiS Preprint Repository

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, proved first by Lorent (for bilipschitz invertible mapping with control of order ${\epsilon }^{\frac{1}{800}}$) then later by by Conti, Schweizer in full generality with optimal control. More specifically we will show that a "quantitative" two well Liouville theorem holds for the set of matrices $K=SO\left(2\right)\cup SO\left(2\right)H$ where $H$ is a diagonal matrix with determinant equal to $1$, under a constraint on the $L^p$ norm of the second derivative. Our theorem is the following.

Let $p\ge 1$, $q>1$. Let $u\in W^{2,p}\left(B_1\left(0\right)\right)\cap W^{1,q}\left(B_1\left(0\right)\right)$. There exists positive constants ${\mathcal{C}}_1<1,{\mathcal{C}}_2>>1$ depending only on $\sigma$, $p$, $q$ such that if $u$ satisfies the following inequalities $${\int }_{B_1\left(0\right)}{d}^q(Du\left(z\right),K)d{L}^{2}z\le {\mathcal{C}}_1\epsilon ,{\int }_{B_1\left(0\right)}{\left|D^{2}u\left(z\right)\right|}^{p}d{L}^{2}z\le {\mathcal{C}}_1{\epsilon }^{1-p}$$ then there exist $A\in K$ such that $${\int }_{B_{\frac{1}{2}}\left(0\right)}{|Du\left(z\right)-A|}^{q}d{L}^{2}z\le {\mathcal{C}}_2{\epsilon }^{\frac{1}{2q}}.$$ We provide a proof of this result by use of a theorem related to the isoperimetric inequality, the approach is conceptually simpler than those previously used, however it does not given the optimal $c{\epsilon }^{\frac{1}{q}}$ bound for ($\text{???}$) that has been proved (for the $p=1$ case) by Conti, Schweizer.