A Two Well Liouville Theorem

Andrew Lorent

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

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

Abstract

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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 = σ 0 0 σ - 1 for σ > 0 . Let 0 < ζ 1 < 1 < ζ 2 < . Let K : = S O 2 S O 2 H . Let u W 2 , 1 Q 1 0 be a invertible bilipschitz function with Lip u < ζ 2 , Lip u - 1 < ζ 1 - 1 . 
There exists positive constants 𝔠 1 < 1 and 𝔠 2 > 1 depending only on σ, ζ 1 , ζ 2 such that if ϵ 0 , 𝔠 1 and u satisfies the following inequalities Q 1 0 d D u z , K d L 2 z ϵ Q 1 0 D 2 u z d L 2 z 𝔠 1 , then there exists J I d , H and R S O 2 such that Q 𝔠 1 0 D u z - R J d L 2 z 𝔠 2 ϵ 1 800 .

How to cite

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Lorent, 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 -

References

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