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 (2009)

  • 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 A F denote the subspace of functions in W 2 , 2 Ω that satisfy the affine boundary condition D u = 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 = l F on ∂ Ω (where l F 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 (2009): 322-366. <http://eudml.org/doc/245484>.

@article{Lorent2009,
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 $A_F$ 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 $\epsilon $) 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 $v=l_F$ on $\partial \Omega $ (where $l_F$ 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 $\mathcal \{C\}_1&lt;1&lt;\mathcal \{C\}_2$ (depending on $A_1,\dots , A_\{N\}$, $p$) for which the following holds true\[ \mathcal \{C\}\_1\alpha \_p\left(\sqrt\{\epsilon \}\right)\le m^p\_\{\epsilon \}\le \mathcal \{C\}\_2\alpha \_p\left(\sqrt\{\epsilon \}\right) \text\{ for all \}\epsilon &gt;0. \]},
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},
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/245484},
volume = {15},
year = {2009},
}

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
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 $A_F$ 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 $\epsilon $) 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 $v=l_F$ on $\partial \Omega $ (where $l_F$ 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 $\mathcal {C}_1&lt;1&lt;\mathcal {C}_2$ (depending on $A_1,\dots , A_{N}$, $p$) for which the following holds true\[ \mathcal {C}_1\alpha _p\left(\sqrt{\epsilon }\right)\le m^p_{\epsilon }\le \mathcal {C}_2\alpha _p\left(\sqrt{\epsilon }\right) \text{ for all }\epsilon &gt;0. \]
LA - eng
KW - two wells; surface energy; non-convex variational problem; crystalline microstructure; stored energy density function; quasiconvex hull
UR - http://eudml.org/doc/245484
ER -

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