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

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ESAIM: COCV
DOI: 10.1051/cocv:2008039

The regularisation of the N-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions

Andrew Lorent

MIS MPG, Inselstrasse 22, 04103 Leipzig, Germany. andrew.lorent@sns.it

Received June 7, 2007. Revised January 15, 2008. Published online June 24, 2008.

Abstract
Let $K:=SO\left(2\right)A_1\cup SO\left(2\right)A_2\dots SO\left(2\right)A_{N}$ where $A_1,A_2,\dots, A_{N}$ are matrices of non-zero determinant. We establish a sharp relation between the following two minimisation problems in two dimensions. Firstly the N-well problem with surface energy. Let $p\in\left[1,2\right]$ , $\Omega\subset \mathbb{R} ^2$ be a convex polytopal region. Define

$\begin{displaymath}I^p_{\epsilon}\left(u\right)=\int_{\Omega} d^p\left(Du\left(z... ...psilon\left\vert D^2 u\left(z\right)\right\vert^2 {\rm d}L^2 z \end{displaymath}$

and let A_F denote the subspace of functions in $W^\left(\Omega\right)$ that satisfy the affine boundary condition Du = F on $\partial \Omega$ (in the sense of trace), where $F\not\in K$ . We consider the scaling (with respect to $\epsilon$ ) of

$\begin{displaymath}m^p_{\epsilon}:=\inf_{u\in A_F} I^p_{\epsilon}\left(u\right). \end{displaymath}$

Secondly the finite element approximation to the N-well problem without surface energy. We will show there exists a space of functions $\mathcal$ where each function $v\in \mathcal$ is piecewise affine on a regular (non-degenerate) h-triangulation and satisfies the affine boundary condition v = l_F on $\partial \Omega$ (where l_F is affine with Dl_F = F) such that for

$\begin{displaymath}\alpha_p\left(h\right):=\inf_{v\in \mathcal} \int_{\Omega}d^p\left(Dv\left(z\right),K\right) {\rm d}L^2 z \end{displaymath}$

there exists positive constants $\mathcal{C}_1<1<\mathcal{C}_2$ (depending on $A_1,\dots, A_{N}$ , p) for which the following holds true

$\begin{displaymath}\mathcal{C}_1\alpha_p\left(\sqrt{\epsilon}\right)\leq m^p_{\e... ...lpha_p\left(\sqrt{\epsilon}\right) \text{ for all }\epsilon>0. \end{displaymath}$