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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

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