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 $p\in \left[1,2\right]$, $\Omega \subset {\mathbb{R}}^{2}$ be a convex polytopal region. Define

$${I}_{\u03f5}^{p}\left(u\right)={\int}_{\Omega}{d}^{p}\left(Du\left(z\right),K\right)+\u03f5{\left|{D}^{2}u\left(z\right)\right|}^{2}\mathrm{d}{L}^{2}z$$ |

$${m}_{\u03f5}^{p}:=\underset{u\in {A}_{F}}{inf}{I}_{\u03f5}^{p}\left(u\right).$$ |

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

$${\mathcal{C}}_{1}{\alpha}_{p}\left(\sqrt{\u03f5}\right)\le {m}_{\u03f5}^{p}\le {\mathcal{C}}_{2}{\alpha}_{p}\left(\sqrt{\u03f5}\right)\phantom{\rule{4pt}{0ex}}\text{for}\phantom{\rule{4pt}{0ex}}\text{all}\phantom{\rule{4pt}{0ex}}\u03f5>0.$$ |

@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}, mrnumber = {2513089}, zbl = {1161.74044}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2008039/} }

TY - JOUR AU - Lorent, Andrew TI - The regularisation of the $N$-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2009 SP - 322 EP - 366 VL - 15 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2008039/ DO - 10.1051/cocv:2008039 LA - en ID - COCV_2009__15_2_322_0 ER -

%0 Journal Article %A Lorent, Andrew %T The regularisation of the $N$-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions %J ESAIM: Control, Optimisation and Calculus of Variations %D 2009 %P 322-366 %V 15 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2008039/ %R 10.1051/cocv:2008039 %G en %F 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, Volume 15 (2009) no. 2, pp. 322-366. doi : 10.1051/cocv:2008039. http://www.numdam.org/articles/10.1051/cocv:2008039/

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