# 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 (2008)

- Volume: 15, Issue: 2, page 322-366
- ISSN: 1292-8119

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topLorent, 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 15.2 (2008): 322-366. <http://eudml.org/doc/90916>.

@article{Lorent2008,

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
$$
I^p\_\{\epsilon\}\left(u\right)=\int\_\{\Omega\} d^p\left(Du\left(z\right),K\right)+\epsilon\left|D^2
u\left(z\right)\right|^2 \{\rm d\}L^2 z
$$
and let AF 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 $F\not\in
K$. We consider the scaling (with respect to ϵ) of
$$
m^p\_\{\epsilon\}:=\inf\_\{u\in A\_F\} I^p\_\{\epsilon\}\left(u\right).
$$
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=lF on $\partial \Omega$ (where lF is affine with
$Dl_F=F$) such that for
$$
\alpha\_p\left(h\right):=\inf\_\{v\in \mathcal\{D\}\_F^\{h\}\}
\int\_\{\Omega\}d^p\left(Dv\left(z\right),K\right) \{\rm d\}L^2 z
$$
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
$$
\mathcal\{C\}\_1\alpha\_p\left(\sqrt\{\epsilon\}\right)\leq m^p\_\{\epsilon\}\leq
\mathcal\{C\}\_2\alpha\_p\left(\sqrt\{\epsilon\}\right) \text\{ for all \}\epsilon>0.
$$},

author = {Lorent, Andrew},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Two wells; surface energy; non-convex variational problem; crystalline microstructure; stored energy density function; quasiconvex hull},

language = {eng},

month = {6},

number = {2},

pages = {322-366},

publisher = {EDP Sciences},

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

url = {http://eudml.org/doc/90916},

volume = {15},

year = {2008},

}

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

DA - 2008/6//

PB - EDP Sciences

VL - 15

IS - 2

SP - 322

EP - 366

AB -
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
$$
I^p_{\epsilon}\left(u\right)=\int_{\Omega} d^p\left(Du\left(z\right),K\right)+\epsilon\left|D^2
u\left(z\right)\right|^2 {\rm d}L^2 z
$$
and let AF 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 $F\not\in
K$. We consider the scaling (with respect to ϵ) of
$$
m^p_{\epsilon}:=\inf_{u\in A_F} I^p_{\epsilon}\left(u\right).
$$
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=lF on $\partial \Omega$ (where lF is affine with
$Dl_F=F$) such that for
$$
\alpha_p\left(h\right):=\inf_{v\in \mathcal{D}_F^{h}}
\int_{\Omega}d^p\left(Dv\left(z\right),K\right) {\rm d}L^2 z
$$
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
$$
\mathcal{C}_1\alpha_p\left(\sqrt{\epsilon}\right)\leq m^p_{\epsilon}\leq
\mathcal{C}_2\alpha_p\left(\sqrt{\epsilon}\right) \text{ for all }\epsilon>0.
$$

LA - eng

KW - Two wells; surface energy; non-convex variational problem; crystalline microstructure; stored energy density function; quasiconvex hull

UR - http://eudml.org/doc/90916

ER -

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