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MiS Preprint

21/2004

A Two Well Liouville Theorem

Andrew Lorent

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 $H=diag\left(\sigma,\sigma^{-1}\right)$ be a $2\times 2$ diagonal matrix. Let $0<\zeta_1<1<\zeta_2<\infty$. Let $K:=SO\left(2\right)\cup SO\left(2\right)H$. Let $u\in W^{2,1}\left(Q_{1}\left(0\right)\right)$ be a $C^1$ invertible bilipschitz function with $\mathrm{Lip}\left(u\right)<\zeta_2$, $\mathrm{Lip}\left(u^{-1}\right)<\zeta_1^{-1}$.

There exists positive constants $c_11$ and $c_2>1$ depending only on $\sigma$, $\zeta_1$, $\zeta_2$ such that if $\epsilon\in\left(0,c_1\right)$ and $u$ satisfies the following inequalities $$ \int_{Q_{1}\left(0\right)} d\left(Du\left(z\right),K\right) dL^2 z\leq \epsilon $$ $$ \int_{Q_{1}\left(0\right)} \left|D^2 u\left(z\right)\right| dL^2 z\leq c_1, $$ then there exists $J\in\left\{Id,H\right\}$ and $R\in SO\left(2\right)$ such that $$ \int_{Q_{c_1}\left(0\right)} \left|Du\left(z\right)-RJ\right| dL^2 z\leq c_2\epsilon^{\frac{1}{800}}. $$