Search

MiS Preprint Repository

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
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}.

Received:
Apr 22, 2004
Published:
Apr 22, 2004
MSC Codes:
74G65
Keywords:
two wells, surface energy

Related publications

inJournal
2006 Repository Open Access
Andrew Lorent

The two-well problem with surface energy

In: Proceedings of the Royal Society of Edinburgh / A, 136 (2006) 4, pp. 795-805