MiS Preprint Repository

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 ${ϵ}^{\frac{1}{3q}}$ 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.