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MiS Preprint
65/2005
On the scaling of the two well problem
Andrew Lorent
Abstract
We establish a sharp relation between the two well problem with surface energy and the finite element approximation to a version of the two well problem.
We will show that if the finite element approximation has a lower bound of $h^{\frac{1}{3}}$ then this implies lower bounds of $\epsilon^{\frac{1}{3q}}$ for the two well problem with surface energy term given by the $L^q$ norm of the second derivative.
Our main tool for establishing this is an $L^q$ two well (suboptimal) Liouville theorem, which we will provide a simple proof of using the case of equality in the isoperimetric inequality. Using the optimal $L^1$ two well Liouville theorem of Conti Schweizer we give a cleaner formulation of our result for the $q=1$ case.
