On the scaling of the two well problem

Andrew Lorent
(Please use for correspondence this email).

Submission date: 22. Jun. 2005
Pages: 33
paper submitted to: Calculus of variations and partial differential equations
MSC-Numbers: 74N15
Keywords and phrases: two wells, surface energy
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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 then this implies lower bounds of for the two well problem with surface energy term given by the norm of the second derivative.

Our main tool for establishing this is an 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 two well Liouville theorem of Conti Schweizer we give a cleaner formulation of our result for the q=1 case.