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

Andrew Lorent

ESAIM: Control, Optimisation and Calculus of Variations (2008)

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

Abstract

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Let K : = S O 2 A 1 ∪ S O 2 A 2 ⋯ S O 2 A N where A 1 , A 2 , ⋯ , 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 ∈ 1 , 2 , Ω ⊂ ℝ 2 be a convex polytopal region. Define I ϵ p u = ∫ Ω d p D u z , K + ϵ D 2 u z 2 d L 2 z and let AF denote the subspace of functions in W 2 , 2 Ω that satisfy the affine boundary condition Du=F on ∂ Ω (in the sense of trace), where F ∉ K . We consider the scaling (with respect to ϵ) of m ϵ p : = inf u ∈ A F I ϵ p u . 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 ∈ 𝒟 F h is piecewise affine on a regular (non-degenerate) h-triangulation and satisfies the affine boundary condition v=lF on ∂ Ω (where lF is affine with D l F = F ) such that for α p h : = inf v ∈ 𝒟 F h ∫ Ω d p D v z , K d L 2 z there exists positive constants 𝒞 1 < 1 < 𝒞 2 (depending on A 1 , ⋯ , A N , p) for which the following holds true 𝒞 1 α p ϵ ≤ m ϵ p ≤ 𝒞 2 α p ϵ for all ϵ > 0 .

How to cite

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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 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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