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 $H=\left({\textstyle \begin{array}{cc}\sigma & 0\\ 0& {\sigma}^{-1}\end{array}}\right)$ 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 ${\U0001d520}_{1}<1$ and ${\U0001d520}_{2}>1$ depending only on $\sigma $, ${\zeta}_{1}$, ${\zeta}_{2}$ such that if $\u03f5\in \left(0,{\U0001d520}_{1}\right)$ and $u$ satisfies the following inequalities

$$\phantom{\rule{-56.9055pt}{0ex}}{\int}_{{Q}_{1}\left(0\right)}\mathrm{d}\left(Du\left(z\right),K\right)\mathrm{d}{L}^{2}z\le \u03f5$$ |

$$\phantom{\rule{-56.9055pt}{0ex}}{\int}_{{Q}_{1}\left(0\right)}\left|{D}^{2}u\left(z\right)\right|\mathrm{d}{L}^{2}z\le {\U0001d520}_{1},$$ |

$$\phantom{\rule{-56.9055pt}{0ex}}{\int}_{{Q}_{{\U0001d520}_{1}}\left(0\right)}\left|Du\left(z\right)-RJ\right|\mathrm{d}{L}^{2}z\le {\U0001d520}_{2}{\u03f5}^{\frac{1}{800}}.$$ |

@article{COCV_2005__11_3_310_0, author = {Lorent, Andrew}, title = {A two well {Liouville} theorem}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {310--356}, publisher = {EDP-Sciences}, volume = {11}, number = {3}, year = {2005}, doi = {10.1051/cocv:2005009}, mrnumber = {2148848}, zbl = {1082.74039}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2005009/} }

TY - JOUR AU - Lorent, Andrew TI - A two well Liouville theorem JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2005 SP - 310 EP - 356 VL - 11 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2005009/ DO - 10.1051/cocv:2005009 LA - en ID - COCV_2005__11_3_310_0 ER -

Lorent, Andrew. A two well Liouville theorem. ESAIM: Control, Optimisation and Calculus of Variations, Volume 11 (2005) no. 3, pp. 310-356. doi : 10.1051/cocv:2005009. http://archive.numdam.org/articles/10.1051/cocv:2005009/

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