A two well Liouville theorem

Lorent, Andrew

ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 3, pp. 310-356.

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 $H=\left(\begin{array}{cc}\sigma & 0\\ 0& {\sigma }^{-1}\end{array}\right)$ for $\sigma >0$. 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 ${\mathrm{C}}^1$ invertible bilipschitz function with $\mathrm{Lip}\left(u\right)<{\zeta }_2$, $\mathrm{Lip}\left(u^{-1}\right)<{\zeta }_1^{-1}$. There exists positive constants ${𝔠}_1<1$ and ${𝔠}_2>1$ depending only on $\sigma$, ${\zeta }_1$, ${\zeta }_2$ such that if $ϵ\in \left(0,{𝔠}_1\right)$ and $u$ satisfies the following inequalities

$${\int }_{Q_1\left(0\right)}\mathrm{d}\left(Du\left(z\right),K\right)\mathrm{d}{L}^{2}z\le ϵ$$

$${\int }_{Q_1\left(0\right)}\left|D^{2}u\left(z\right)\right|\mathrm{d}{L}^{2}z\le {𝔠}_1,$$

then there exists $J\in \left\{Id,H\right\}$ and $R\in SO\left(2\right)$ such that

$${\int }_{Q_{{𝔠}_1}\left(0\right)}\left|Du\left(z\right)-RJ\right|\mathrm{d}{L}^{2}z\le {𝔠}_2{ϵ}^{\frac{1}{800}}.$$

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