A Two Well Liouville Theorem

Andrew Lorent

doi:10.1051/cocv:2005009

All issues

Volume 11 / No 3 (July 2005)

ESAIM: COCV, 11 3 (2005) 310-356

Abstract

Free Access

Issue		ESAIM: COCV Volume 11, Number 3, July 2005


Page(s)		310 - 356
DOI		https://doi.org/10.1051/cocv:2005009
Published online		15 July 2005

ESAIM: COCV 11 (2005) 310-356

A Two Well Liouville Theorem

Andrew Lorent

Mathematical Institute, 24-29 St Giles', Oxford, UK; lorent@maths.ox.ac.uk

Received: 21 April 2004

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 $\int_{Q_{\mathfrak{c}_1}\left(0\right)} \left|Du\left(z\right)-RJ\right| {\rm d}L^2 z\leq \mathfrak{c}_2\epsilon^{\frac{1}{800}}.$

Mathematics Subject Classification: 74N15

Key words: Two wells / Liouville.

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