MiS Preprint Repository

Let $\Omega$ be a bounded domain in $\mathbb{R}^2$, let $H$ be a $2\times 2$ matrix with $\mathrm{det}\left(H\right)=1$. Let $\epsilon>0$ and consider the functional $I_{\epsilon}\left(u\right):=\int_{\Omega} \mathrm{dist}\left(Du\left(z\right),SO\left(2\right)\cup SO\left(2\right)H\right)+\epsilon\left|D^2 u\left(z\right)\right| dL^2 z$ over the class $\mathcal{B}_{F}$ of Lipschitz functions from $\Omega$ satisfying affine boundary condition $F$. It can be shown by convex integration that there exists $F\not\in SO\left(2\right)\cup SO\left(2\right)H$ and $u\in\mathcal{B}_F$ with $I_{0}\left(u\right)=0$. In this paper we begin the study of the asymptotics of $m_{\epsilon}:=\inf_{\mathcal{B}_F\cap W^{2,1}} I_{\epsilon}$ 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 $\lim\inf _{\epsilon\rightarrow 0} \frac{m_{\epsilon}}{\epsilon}=\infty$.

In this paper we link the behavior of $m_{\epsilon}$ to the minimum of $I_{0}$ over a suitable class of piecewise affine functions. Let $\left\{\tau_i\right\}$ be a triangulation of $\Omega$ 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 $$ \frac{I_{\epsilon}\left(u\right)}{\epsilon}\leq \epsilon^{-\beta} $$ then for $h\approx \epsilon^{\sqrt{\beta}}$ the interpolant $\tilde{u}\in A^h_F$ satisfies $I_0\left(\tilde{u}\right)\leq h^{1-c\sqrt{\beta}}$.

Note that it is conjectured that $\inf_{v\in A^h_F} I_0\left(v\right)\approx h^{\frac{1}{3}}$ and it is trivial that $\inf_{v\in A^h_F} I_0\left(v\right)\geq c_0 h$ so we reduce the problem of non-trivial lower bounds on $\inf_{\mathcal{B}_F\cap C^2\cap\mathrm{Bilip}} \frac{I_{\epsilon}}{\epsilon}$ to the problem of non-trivial lower bounds on $\inf_{v\in A^h_F} I_0$. This latter point will be addressed in a forthcoming paper.