Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.
MiS Preprint
37/2002
Lower bounds for the two well problem with surface energy I: Reduction to finite elements
Andrew Lorent
Abstract
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.