An optimal scaling law for finite element approximations of a variational problem with non-trivial microstructure

Andrew Lorent

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A two well Liouville theorem

Andrew Lorent (2005)

ESAIM: Control, Optimisation and Calculus of Variations

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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 $H = (\begin{matrix} σ & 0 \\ 0 & σ^{- 1} \end{matrix})$ for $σ > 0$ . Let $0 < ζ_{1} < 1 < ζ_{2} < \infty$ . Let $K : = S O (2) \cup S O (2) H$ . Let $u \in W^{2, 1} (Q_{1} (0))$ 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 $ϵ \in (0, 𝔠_{1})$ and $u$ ...