# Surface energy for the two well problem

##
Andrew Lorent
(University of Cincinnati)

created by alberti on 04 Mar 2003

6 mar 2003

**Abstract.**

Seminari del Giovedi':

speaker:Andrew Lorent
titolo: Surface energy for the two well problem
data: Giovedi' 6 marzo
ora: 15.00
luogo: sala delle riunioni, Centro De Giorgi

Abstract:
We study the functional $I_{\epsilon}$ over a class of functions
from a region $\Omega\subset\R^2$ into $\R^2$ having affine
boundary condition. Functional $I_{\epsilon}(u)$ is a competition
between two terms; the "bulk energy" term that forces the derivatives
of u to be close to $SO(2) \cup SO(2)H$ (where H is a diagonal
matrix) and the "surface energy" term which is given by
$\int_{\Omega} \epsilon

D^2 u

$. $I_{\epsilon}$ is the simplest
functional for which in the case where surface energy is set to zero
(i.e. for $I_{0}$) there exists an exact minimiser via convex integration.
We let $m_{\epsilon}=\inf_{A} I_{\epsilon}$ where $A$ is the
appropriate function class (i.e. having affine boundary condition,
being in Sobolev space) we begin the study of the scaling of
$m_{\epsilon}/\epsilon$ as $\epsilon\rightarrow 0$. This scaling is
important because a full understanding of it will in some sense
answer the question of whether convex integration has an effect on
minimisation problems that in some very small way constrain
oscillation. All that is known is that $m_{\epsilon}/\epsilon\rightarrow
\infty$ as $\epsilon\rightarrow 0$. We will show that the problem of
the scaling of $m_{\epsilon}/\epsilon$ is linked to the problem of the
scaling of $I_{0}$ over finite element spaces (i.e. functions that are
piecewise affine on a triangular grid) We will indicate how this gives
hope an approach to proving scaling lower bounds.