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 
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.