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MiS Preprint
22/2004
The Two Well Problem With Surface Energy
Andrew Lorent
Abstract
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^2$, let $H$ be a $2\times 2$ diagonal 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 $\mathcal{A}_{F}\cap W^{2,1}$ where $\mathcal{A}_F$ is the class of functions from $\Omega$ satisfying affine boundary condition $F$.
We will show that non-trivial (scaling) lower bounds on I_{\ep} follow from non trivial (scaling) lower bounds on the finite element approximation of I_{0}.
