A two well Liouville theorem

Lorent, Andrew. "A two well Liouville theorem." ESAIM: Control, Optimisation and Calculus of Variations 11.3 (2005): 310-356. <http://eudml.org/doc/245991>.

@article{Lorent2005,
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 $H=\bigl (\{\{\textstyle \begin\{matrix\} \sigma & 0\\ 0 & \sigma ^\{-1\} \end\{matrix\}\}\}\bigr )$ for $\sigma &gt;0$. Let $0&lt;\zeta _1&lt;1&lt;\zeta _2&lt;\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 $\mathrm \{Lip\}\left(u\right)&lt;\zeta _2$, $\mathrm \{Lip\}\left(u^\{-1\}\right)&lt;\zeta _1^\{-1\}$. There exists positive constants $\mathfrak \{c\}_1&lt;1$ and $\mathfrak \{c\}_2&gt;1$ depending only on $\sigma $, $\zeta _1$, $\zeta _2$ such that if $\epsilon \in \left(0,\mathfrak \{c\}_1\right)$ and $u$ satisfies the following inequalities\[\hspace*\{-56.9055pt\} \int \_\{Q\_\{1\}\left(0\right)\} \{\rm d\}\left(Du\left(z\right),K\right) \{\rm d\}L^2 z\le \epsilon \]\[\hspace*\{-56.9055pt\} \int \_\{Q\_\{1\}\left(0\right)\} \left|D^2 u\left(z\right)\right| \{\rm d\}L^2 z\le \mathfrak \{c\}\_1, \]then there exists $J\in \left\lbrace Id,H\right\rbrace $ and $R\in SO\left(2\right)$ such that\[\hspace*\{-56.9055pt\} \int \_\{Q\_\{\mathfrak \{c\}\_1\}\left(0\right)\} \left|Du\left(z\right)-RJ\right| \{\rm d\}L^2 z\le \mathfrak \{c\}\_2\epsilon ^\{\frac\{1\}\{800\}\}. \]},
author = {Lorent, Andrew},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {two wells; Liouville; constraint surface energy},
language = {eng},
number = {3},
pages = {310-356},
publisher = {EDP-Sciences},
title = {A two well Liouville theorem},
url = {http://eudml.org/doc/245991},
volume = {11},
year = {2005},
}

TY - JOUR
AU - Lorent, Andrew
TI - A two well Liouville theorem
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2005
PB - EDP-Sciences
VL - 11
IS - 3
SP - 310
EP - 356
AB - 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 $H=\bigl ({{\textstyle \begin{matrix} \sigma & 0\\ 0 & \sigma ^{-1} \end{matrix}}}\bigr )$ for $\sigma &gt;0$. Let $0&lt;\zeta _1&lt;1&lt;\zeta _2&lt;\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 $\mathrm {Lip}\left(u\right)&lt;\zeta _2$, $\mathrm {Lip}\left(u^{-1}\right)&lt;\zeta _1^{-1}$. There exists positive constants $\mathfrak {c}_1&lt;1$ and $\mathfrak {c}_2&gt;1$ depending only on $\sigma $, $\zeta _1$, $\zeta _2$ such that if $\epsilon \in \left(0,\mathfrak {c}_1\right)$ and $u$ satisfies the following inequalities\[\hspace*{-56.9055pt} \int _{Q_{1}\left(0\right)} {\rm d}\left(Du\left(z\right),K\right) {\rm d}L^2 z\le \epsilon \]\[\hspace*{-56.9055pt} \int _{Q_{1}\left(0\right)} \left|D^2 u\left(z\right)\right| {\rm d}L^2 z\le \mathfrak {c}_1, \]then there exists $J\in \left\lbrace Id,H\right\rbrace $ and $R\in SO\left(2\right)$ such that\[\hspace*{-56.9055pt} \int _{Q_{\mathfrak {c}_1}\left(0\right)} \left|Du\left(z\right)-RJ\right| {\rm d}L^2 z\le \mathfrak {c}_2\epsilon ^{\frac{1}{800}}. \]
LA - eng
KW - two wells; Liouville; constraint surface energy
UR - http://eudml.org/doc/245991
ER -