MiS Preprint Repository

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.

Firstly the $N$-well problem with surface energy. Let $p\in [1,2]$. Let $$I_{ϵ}^p\left(u\right)={\int }_{\mathrm{\Omega }}{d}^p(Du\left(z\right),K)+ϵ{\left|D^{2}u\left(z\right)\right|}^{2}d{L}^{2}z$$ and let $A_F$ denote the subspace of functions in $W^{2,2}\left(\mathrm{\Omega }\right)$ that satisfy the affine boundary condition $Du=F$ on $\partial \mathrm{\Omega }$ (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 ${\mathcal{D}}_F^h$ where each function $v\in {\mathcal{D}}_F^h$ is piecewise affine on a regular (non-degenerate) $h$-triangulation and satisfies the affine boundary condition $v=l_F$ on $\partial \mathrm{\Omega }$ (where $l_F$ is affine with $D{l}_F=F$) such that for $${\alpha }_p\left(h\right):=\underset{v\in {\mathcal{D}}_F^h}{inf}{\int }_{\mathrm{\Omega }}{d}^p(Dv\left(z\right),K)d{L}^{2}z$$ there exists positive constants ${\mathcal{C}}_1<1<{\mathcal{C}}_2$ (depending on $A_1,\dots A_N$, $\varsigma$, $p$) for which the following holds true $${\mathcal{C}}_1{\alpha }_p\left(\sqrt{ϵ}\right)\le m_{ϵ}^p\le {\mathcal{C}}_2{\alpha }_p\left(\sqrt{ϵ}\right)\text{for}\text{all}ϵ>0.$$