Preprint 65/2005

On the scaling of the two well problem

Andrew Lorent

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Submission date: 22. Jun. 2005
Pages: 33
Bibtex
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 formula13 then this implies lower bounds of formula15 for the two well problem with surface energy term given by the formula17 norm of the second derivative.

Our main tool for establishing this is an formula17 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 formula21 two well Liouville theorem of Conti Schweizer we give a cleaner formulation of our result for the q=1 case.

07.06.2018, 02:11