The regularisation of the $N$-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 2, pp. 322-366.

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
 ${𝒞}_{1}{\alpha }_{p}\left(\sqrt{ϵ}\right)\le {m}_{ϵ}^{p}\le {𝒞}_{2}{\alpha }_{p}\left(\sqrt{ϵ}\right)\phantom{\rule{4pt}{0ex}}\text{for}\phantom{\rule{4pt}{0ex}}\text{all}\phantom{\rule{4pt}{0ex}}ϵ>0.$

DOI : https://doi.org/10.1051/cocv:2008039
Classification : 74N15
Mots clés : two wells, surface energy
@article{COCV_2009__15_2_322_0,
author = {Lorent, Andrew},
title = {The regularisation of the $N$-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {322--366},
publisher = {EDP-Sciences},
volume = {15},
number = {2},
year = {2009},
doi = {10.1051/cocv:2008039},
zbl = {1161.74044},
mrnumber = {2513089},
language = {en},
url = {http://www.numdam.org/item/COCV_2009__15_2_322_0/}
}
Lorent, Andrew. The regularisation of the $N$-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 2, pp. 322-366. doi : 10.1051/cocv:2008039. http://www.numdam.org/item/COCV_2009__15_2_322_0/

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