An Lp two well Liouville Theorem
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Submission date: 14. Aug. 2006 (revised version: August 2006)
published in: Annales Academiae Scientiarum Fennicae / Mathematica, 33 (2008) 2, p. 439-473
Keywords and phrases: two wells, Liouviille
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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.