Calculus of Variations and Geometric Measure Theory
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Surface energy for the two well problem

Andrew Lorent (University of Cincinnati)

created by alberti on 04 Mar 2003

6 mar 2003


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.

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