A two well Liouville theorem
ESAIM: Control, Optimisation and Calculus of Variations, Volume 11 (2005) no. 3, pp. 310-356.

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=σ00σ -1 for σ>0. Let 0<ζ 1 <1<ζ 2 <. Let K:=SO2SO2H. Let uW 2,1 Q 1 0 be a C 1 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 ϵ0,𝔠 1 and u satisfies the following inequalities

Q 1 0 dDuz,KdL 2 zϵ
Q 1 0 D 2 uzdL 2 z𝔠 1 ,
then there exists JId,H and RSO2 such that
Q 𝔠 1 0 Duz-RJdL 2 z𝔠 2 ϵ 1 800 .

DOI: 10.1051/cocv:2005009
Classification: 74N15
Keywords: two wells, Liouville
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Lorent, Andrew. A two well Liouville theorem. ESAIM: Control, Optimisation and Calculus of Variations, Volume 11 (2005) no. 3, pp. 310-356. doi : 10.1051/cocv:2005009. http://www.numdam.org/articles/10.1051/cocv:2005009/

[1] 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). | MR | Zbl

[2] J.M. Ball and R.D. James, Fine phase mixtures as minimisers of energy. Arch. Rat. Mech. Anal. 100 (1987) 13-52. | Zbl

[3] 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. A 338 (1992) 389-450. | Zbl

[4] N. Chaudhuri and S. Müller, Rigidity Estimate for Two Incompatible Wells. Calc. Var. Partial Differ. Equ. 19 (2004) 379-390. | Zbl

[5] M. Chipot and D. Kinderlehrer, Equilibrium configurations of crystals. Arch. Rat. Mech. Anal. 103 (1988) 237-277. | Zbl

[6] 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.

[7] S. Conti, D. Faraco and F. Maggi, A new approach to counterexamples to L 1 estimates: Korn’s inequality, geometric rigidity, and regularity for gradients of separately convex functions. Arch. Rat. Mech. Anal. 175 (2005) 287-300. | Zbl

[8] S. Conti and B. Schweizer, A sharp-interface limit for a two-well problem in geometrically linear elasticity. MPI MIS Preprint Nr. 87/2003. | MR | Zbl

[9] S. Conti and B. Schweizer, Rigidity and Gamma convergence for solid-solid phase transitions with SO(2)-invariance. MPI MIS Preprint Nr. 69/2004. | MR

[10] B. Dacorogna and P. Marcellini, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases. Acta Math. 178 (1997) 1-37. | Zbl

[11] 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. | Zbl

[12] A. Lorent, An optimal scaling law for finite element approximations of a variational problem with non-trivial microstructure. ESAIM: M2AN 35 (2001) 921-934. | Numdam | Zbl

[13] A. Lorent, The two well problem with surface energy. MPI MIS Preprint No. 22/2004. | MR | Zbl

[14] A. Lorent, On the scaling of the two well problem. Forthcoming.

[15] 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. | Zbl

[16] 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. | Zbl

[17] O. Pantz, On the justification of the nonlinear inextensional plate model. Arch. Ration. Mech. Anal. 167 (2003) 179-209. | Zbl

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