# 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 -

## References

top- L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Math. Monogr. The Clarendon Press, Oxford University Press, New York (2000).
- J.M. Ball and R.D. James, Fine phase mixtures as minimisers of energy. Arch. Rat. Mech. Anal.100 (1987) 13–52.
- J.M. Ball and R.D. James, Proposed experimental tests of a theory of fine microstructure and the two well problem. Phil. Trans. Roy. Soc. London Ser. A338 (1992) 389–450.
- N. Chaudhuri and S. Müller, Rigidity Estimate for Two Incompatible Wells. Calc. Var. Partial Differ. Equ.19 (2004) 379–390.
- M. Chipot and D. Kinderlehrer, Equilibrium configurations of crystals. Arch. Rat. Mech. Anal.103 (1988) 237–277.
- M. Chipot and S. Müller, Sharp energy estimates for finite element approximations of non-convex problems. Variations of domain and free-boundary problems in solid mechanics (Paris, 1997). Solid Mech. Appl.66 (1999) 317–325.
- S. Conti, D. Faraco and F. Maggi, A new approach to counterexamples to L1 estimates: Korn's inequality, geometric rigidity, and regularity for gradients of separately convex functions. Arch. Rat. Mech. Anal.175 (2005) 287–300.
- S. Conti and B. Schweizer, A sharp-interface limit for a two-well problem in geometrically linear elasticity. MPI MIS Preprint Nr. 87/2003.
- S. Conti and B. Schweizer, Rigidity and Gamma convergence for solid-solid phase transitions with$SO\left(2\right)$-invariance. MPI MIS Preprint Nr. 69/2004.
- B. Dacorogna and P. Marcellini, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases. Acta Math.178 (1997) 1–37.
- G. Friesecke, R.D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity. Comm. Pure Appl. Math.55 (2002) 1461–1506.
- A. Lorent, An optimal scaling law for finite element approximations of a variational problem with non-trivial microstructure. ESAIM: M2AN35 (2001) 921–934.
- A. Lorent, The two well problem with surface energy. MPI MIS Preprint No. 22/2004.
- A. Lorent, On the scaling of the two well problem. Forthcoming.
- S. Müller and V. Šverák, Attainment results for the two-well problem by convex integration, in Geometric Analysis and the Calculus of Variations, Stefan Hildebrandt, J. Jost Ed. International Press, Cambridge (1996) 239–251.
- S. Müller and V. Šverák, Convex integration with constraints and applications to phase transitions and partial differential equations. J. Eur. Math. Soc.1 (1999) 393–422.
- O. Pantz, On the justification of the nonlinear inextensional plate model. Arch. Ration. Mech. Anal.167 (2003) 179–209.

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