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MiS Preprint

72/2006

An $L^p$ two well Liouville Theorem

Andrew Lorent

Abstract

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 $\varepsilon^{\frac{1}{800}}$) 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 $p\geq 1$, $q> 1$. 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 $$\label{aa1} \int_{B_{1}\left(0\right)} d^q\left(Du\left(z\right),K\right) dL^2 z\leq \mathcal{C}_1\varepsilon,\;\;\;\int_{B_{1}\left(0\right)} \left|D^2 u\left(z\right)\right|^p dL^2 z\leq \mathcal{C}_1 \varepsilon^{1-p} $$ then there exist $A\in K$ such that \begin{equation} \label{aa3} \int_{B_{\frac{1}{2}}\left(0\right)} \left|Du\left(z\right)-A\right|^q dL^2 z\leq \mathcal{C}_2\varepsilon^{\frac{1}{2q}}. \end{equation} 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 $c\varepsilon^{\frac{1}{q}}$ bound for (\ref{aa3}) that has been proved (for the $p=1$ case) by Conti, Schweizer.