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### An optimal scaling law for finite element approximations of a variational problem with non-trivial microstructure

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this note we give sharp lower bounds for a non-convex functional when minimised over the space of functions that are piecewise affine on a triangular grid and satisfy an affine boundary condition in the second lamination convex hull of the wells of the functional.

### On indecomposable sets with applications

ESAIM: Control, Optimisation and Calculus of Variations

In this note we show the characteristic function of every indecomposable set in the plane is equivalent to the characteristic function a closed set See Formula in PDF See Formula in PDF . We show by example this is false in dimension three and above. As a corollary to this result we show that for every &gt; 0 a set of finite perimeter can be approximated by a closed subset See Formula in PDF See Formula in PDF with finitely many indecomposable components and with the property that See Formula...

### 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

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...

### A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity

ESAIM: Control, Optimisation and Calculus of Variations

The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain  ⊂ ℝthe functional is ${I}_{}\left(u\right)=\frac{1}{2}{\int }_{\Omega }^{-1}{|1-|Du|}^{2}{|}^{2}+{|{D}^{2}u|}^{2}\mathrm{d}z$ I ϵ ( u ) = 1 2 ∫ Ω ϵ -1 1 − Du 2 2 + ϵ D 2 u 2 d z wherebelongs to the subset of functions in ${W}_{0}^{2,2}\left(\Omega \right)$W02,2(Ω) whose gradient (in the sense of trace) satisfies()·  = 1 where is the inward pointing unit normal to at . In [1 (2002) 187–202]...

### A two well Liouville theorem

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller. Let $H=\left(\begin{array}{cc}\sigma & 0\\ 0& {\sigma }^{-1}\end{array}\right)$ for $\sigma >0$. Let $0<{\zeta }_{1}<1<{\zeta }_{2}<\infty$. Let $K:=SO\left(2\right)\cup SO\left(2\right)H$. Let $u\in {W}^{2,1}\left({Q}_{1}\left(0\right)\right)$ be a ${\mathrm{C}}^{1}$ invertible bilipschitz function with $\mathrm{Lip}\left(u\right)<{\zeta }_{2}$, $\mathrm{Lip}\left({u}^{-1}\right)<{\zeta }_{1}^{-1}$. There exists positive constants ${𝔠}_{1}<1$ and ${𝔠}_{2}>1$ depending only on $\sigma$, ${\zeta }_{1}$, ${\zeta }_{2}$ such that if $ϵ\in \left(0,{𝔠}_{1}\right)$ and $u$ satisfies the...

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

ESAIM: Control, Optimisation and Calculus of Variations

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 -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 denote the subspace of functions in ${W}^{2,2}\left(\Omega \right)$ that satisfy the affine boundary condition on $\partial \Omega$ (in the sense of trace), where $F\notin K$. We consider the scaling (with respect to ) of ${m}_{ϵ}^{p}:=\underset{u\in {A}_{F}}{inf}{I}_{ϵ}^{p}\left(u\right).$ Secondly the finite element approximation to the -well problem without...

### A Two Well Liouville Theorem

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller. Let $H=\left(\begin{array}{ccc}\sigma & 00& {\sigma }^{-1}\end{array}\right)$ for $\sigma >0$. Let $0<{\zeta }_{1}<1<{\zeta }_{2}<\infty$. Let $K:=SO\left(2\right)\cup SO\left(2\right)H$. Let $u\in {W}^{2,1}\left({Q}_{1}\left(0\right)\right)$ be a  invertible bilipschitz function with $\mathrm{Lip}\left(u\right)<{\zeta }_{2}$, $\mathrm{Lip}\left({u}^{-1}\right)<{\zeta }_{1}^{-1}$.  There exists positive constants ${𝔠}_{1}<1$ and ${𝔠}_{2}>1$ depending only on , ${\zeta }_{1}$, ${\zeta }_{2}$ such that if $ϵ\in \left(0,{𝔠}_{1}\right)$ and satisfies...

### An optimal scaling law for finite element approximations of a variational problem with non-trivial microstructure

ESAIM: Mathematical Modelling and Numerical Analysis

In this note we give sharp lower bounds for a non-convex functional when minimised over the space of functions that are piecewise affine on a triangular grid and satisfy an affine boundary condition in the second lamination convex hull of the wells of the functional.

### A simple proof of the characterization of functions of low Aviles Giga energy on a ball regularity

ESAIM: Control, Optimisation and Calculus of Variations

The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain  ⊂ ℝ the functional is ${\mathit{I}}_{\mathit{ϵ}}\mathrm{\left(}\mathit{u}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}{{}^{\mathrm{\int }}}_{\mathit{\Omega }}{\mathit{ϵ}}^{-1}{\left|\mathrm{1}\mathrm{-}{\left|\mathit{Du}\right|}^{\mathrm{2}}\right|}^{\mathrm{2}}\mathrm{+}\mathit{ϵ}{\left|{\mathit{D}}^{\mathrm{2}}\mathit{u}\right|}^{\mathrm{2}}\mathrm{d}\mathit{z}$ where belongs to the subset of functions in ${\mathit{W}}_{\mathrm{0}}^{\mathrm{2}\mathit{,}\mathrm{2}}\mathrm{\left(}\mathit{\Omega }\mathrm{\right)}$ whose gradient (in the sense of trace) satisfies ()·  = 1 where is the inward...