MiS Preprint Repository

Let $\mathrm{\Omega }$ be a bounded domain in ${\mathbb{R}}^2$, let $H$ be a $2\times 2$ matrix with $\det \left(H\right)=1$. Let $ϵ>0$ and consider the functional $I_{ϵ}\left(u\right):={\int }_{\mathrm{\Omega }}\mathrm{dist}(Du\left(z\right),SO\left(2\right)\cup SO\left(2\right)H)+ϵ\left|D^{2}u\left(z\right)\right|d{L}^{2}z$ over the class ${\mathcal{B}}_F$ of Lipschitz functions from $\mathrm{\Omega }$ satisfying affine boundary condition $F$. It can be shown by convex integration that there exists $F\notin SO\left(2\right)\cup SO\left(2\right)H$ and $u\in {\mathcal{B}}_F$ with $I_0\left(u\right)=0$. In this paper we begin the study of the asymptotics of $m_{ϵ}:=\underset{{\mathcal{B}}_F\cap W^{2,1}}{inf}{I}_{ϵ}$ for such $F$. This is the simplest minimisation problem involving surface energy in which we can hope to see the effects of convex integration solutions. The only known lower bounds are $lim\underset{ϵ\to 0}{inf}\frac{m_{ϵ}}{ϵ}=\mathrm{\infty }$.

In this paper we link the behavior of $m_{ϵ}$ to the minimum of $I_0$ over a suitable class of piecewise affine functions. Let $\left\{{\tau }_i\right\}$ be a triangulation of $\mathrm{\Omega }$ by triangles of diameter less than $h$ and let $A_F^h$ denote the class of continuous functions that are piecewise affine on a triangulation $\left\{{\tau }_i\right\}$. For function $u\in {\mathcal{A}}_F$ let $\tilde{u}\in A_F^h$ be the interpolant, i.e. the function we obtain by defining ${\tilde{u}}_{\lfloor {\tau }_i}$ to be the affine interpolation of $u$ on the corners of ${\tau }_i$. We show that if for some small $\beta >0$ there exists $u\in {\mathcal{B}}_F\cap C^2\cap \mathrm{Bilip}$ with $$\frac{I_{ϵ}\left(u\right)}{ϵ}\le {ϵ}^{-\beta }$$ then for $h\approx {ϵ}^{\sqrt{\beta }}$ the interpolant $\tilde{u}\in A_F^h$ satisfies $I_0\left(\tilde{u}\right)\le h^{1-c\sqrt{\beta }}$.

Note that it is conjectured that $\underset{v\in A_F^h}{inf}{I}_0\left(v\right)\approx h^{\frac{1}{3}}$ and it is trivial that $\underset{v\in A_F^h}{inf}{I}_0\left(v\right)\ge c_{0}h$ so we reduce the problem of non-trivial lower bounds on $\underset{{\mathcal{B}}_F\cap C^2\cap \mathrm{Bilip}}{inf}\frac{I_{ϵ}}{ϵ}$ to the problem of non-trivial lower bounds on $\underset{v\in A_F^h}{inf}{I}_0$. This latter point will be addressed in a forthcoming paper.