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

Let $K:=SO\left(2\right)A_1\cup SO\left(2\right)A_2\cdots SO\left(2\right)A_N$ where $A_1,A_2,\cdots ,A_N$ 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 $p\in \left[1,2\right]$, $\Omega \subset {ℝ}^2$ be a convex polytopal region. Define

$$I_{ϵ}^p\left(u\right)={\int }_{\Omega }{d}^p\left(Du\left(z\right),K\right)+ϵ{\left|D^{2}u\left(z\right)\right|}^2\mathrm{d}{L}^{2}z$$

and let $A_F$ denote the subspace of functions in $W^{2,2}\left(\Omega \right)$ that satisfy the affine boundary condition $Du=F$ on $\partial \Omega$ (in the sense of trace), where $F\notin K$. We consider the scaling (with respect to $ϵ$) of

$$m_{ϵ}^p:=\underset{u\in A_F}{inf}{I}_{ϵ}^p\left(u\right).$$

Secondly the finite element approximation to the $N$-well problem without surface energy. We will show there exists a space of functions ${𝒟}_F^h$ where each function $v\in {𝒟}_F^h$ 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 $D{l}_F=F$) such that for

$${\alpha }_p\left(h\right):=\underset{v\in {𝒟}_F^h}{inf}{\int }_{\Omega }{d}^p\left(Dv\left(z\right),K\right)\mathrm{d}{L}^{2}z$$

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

$${𝒞}_1{\alpha }_p\left(\sqrt{ϵ}\right)\le m_{ϵ}^p\le {𝒞}_2{\alpha }_p\left(\sqrt{ϵ}\right)\text{for}\text{all}ϵ>0.$$