

Preprint 72/2006
An Lp two well Liouville Theorem
Andrew Lorent
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Submission date: 14. Aug. 2006 (revised version: August 2006)
Pages: 36
published in: Annales Academiae Scientiarum Fennicae / Mathematica, 33 (2008) 2, p. 439-473
Bibtex
MSC-Numbers: 74N15
Keywords and phrases: two wells, Liouviille
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Abstract:
We provide a different approach to and
prove a (partial) generalisation of a recent theorem on the
structure of low energy solutions of the compatible two well
problem in two dimensions, proved first by Lorent (for bilipschitz invertible
mapping with control of order ) then later by
by Conti, Schweizer in full generality with optimal control. More
specifically we will show that a ``quantitative' two well
Liouville theorem holds for the
set of matrices
where
H is a diagonal matrix with determinant equal to 1,
under a constraint on the
norm of
the second derivative. Our theorem is the following.
Let , q> 1. Let
.
There exists positive constants
depending only on
, p, q
such that if u satisfies the following inequalities
then there exist such that
We provide a proof of this result by use of a theorem related to
the isoperimetric inequality, the approach is conceptually simpler
than those previously used, however
it does not given the optimal bound for (1) that has been proved (for the p=1 case) by Conti, Schweizer.