# A Two Well Liouville Theorem

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 11, Issue: 3, page 310-356
- ISSN: 1292-8119

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topLorent, Andrew. "A Two Well Liouville Theorem." ESAIM: Control, Optimisation and Calculus of Variations 11.3 (2010): 310-356. <http://eudml.org/doc/90767>.

@article{Lorent2010,

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 $H=\bigl(\begin\{smallmatrix\} \sigma& 0 0 & \sigma^\{-1\}
\end\{smallmatrix\}\bigr)$ 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 $\xCone$ invertible bilipschitz
function with $\mathrm\{Lip\}\left(u\right)<\zeta_2$, $\mathrm\{Lip\}\left(u^\{-1\}\right)<\zeta_1^\{-1\}$.
There exists positive constants $\mathfrak\{c\}_1<1$ and $\mathfrak\{c\}_2>1$ depending only on σ, $\zeta_1$,
$\zeta_2$ such that if
$\epsilon\in\left(0,\mathfrak\{c\}_1\right)$ and u satisfies the following inequalities
\[
\int\_\{Q\_\{1\}\left(0\right)\} \{\rm d\}\left(Du\left(z\right),K\right) \{\rm d\}L^2 z\leq \epsilon
\]\[
\int\_\{Q\_\{1\}\left(0\right)\} \left|D^2 u\left(z\right)\right| \{\rm d\}L^2 z\leq \mathfrak\{c\}\_1,
\]
then there exists $J\in\left\\{Id,H\right\\}$ and $R\in SO\left(2\right)$ such that
\[
\int\_\{Q\_\{\mathfrak\{c\}\_1\}\left(0\right)\} \left|Du\left(z\right)-RJ\right| \{\rm d\}L^2 z\leq \mathfrak\{c\}\_2\epsilon^\{\frac\{1\}\{800\}\}.
\]},

author = {Lorent, Andrew},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Two wells; Liouville.; constraint surface energy},

language = {eng},

month = {3},

number = {3},

pages = {310-356},

publisher = {EDP Sciences},

title = {A Two Well Liouville Theorem},

url = {http://eudml.org/doc/90767},

volume = {11},

year = {2010},

}

TY - JOUR

AU - Lorent, Andrew

TI - A Two Well Liouville Theorem

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

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 $H=\bigl(\begin{smallmatrix} \sigma& 0 0 & \sigma^{-1}
\end{smallmatrix}\bigr)$ 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 $\xCone$ invertible bilipschitz
function with $\mathrm{Lip}\left(u\right)<\zeta_2$, $\mathrm{Lip}\left(u^{-1}\right)<\zeta_1^{-1}$.
There exists positive constants $\mathfrak{c}_1<1$ and $\mathfrak{c}_2>1$ depending only on σ, $\zeta_1$,
$\zeta_2$ such that if
$\epsilon\in\left(0,\mathfrak{c}_1\right)$ and u satisfies the following inequalities
\[
\int_{Q_{1}\left(0\right)} {\rm d}\left(Du\left(z\right),K\right) {\rm d}L^2 z\leq \epsilon
\]\[
\int_{Q_{1}\left(0\right)} \left|D^2 u\left(z\right)\right| {\rm d}L^2 z\leq \mathfrak{c}_1,
\]
then there exists $J\in\left\{Id,H\right\}$ and $R\in SO\left(2\right)$ such that
\[
\int_{Q_{\mathfrak{c}_1}\left(0\right)} \left|Du\left(z\right)-RJ\right| {\rm d}L^2 z\leq \mathfrak{c}_2\epsilon^{\frac{1}{800}}.
\]

LA - eng

KW - Two wells; Liouville.; constraint surface energy

UR - http://eudml.org/doc/90767

ER -

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