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MiS Preprint
72/2006

An Lp two well Liouville Theorem

Andrew Lorent

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 ε1800) 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 K=SO(2)SO(2)H where H is a diagonal matrix with determinant equal to 1, under a constraint on the Lp norm of the second derivative. Our theorem is the following.

Let p1, q>1. Let uW2,p(B1(0))W1,q(B1(0)). There exists positive constants C1<1,C2>>1 depending only on σ, p, q such that if u satisfies the following inequalities B1(0)dq(Du(z),K)dL2zC1ε,B1(0)|D2u(z)|pdL2zC1ε1p then there exist AK such that B12(0)|Du(z)A|qdL2zC2ε12q. 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 cε1q bound for (???) that has been proved (for the p=1 case) by Conti, Schweizer.

Received:
14.08.06
Published:
14.08.06
MSC Codes:
74N15
Keywords:
two wells, Liouviille

Related publications

inJournal
2008 Journal Open Access
Andrew Lorent

An L-p two well Liouville theorem

In: Annales Academiae Scientiarum Fennicae / Mathematica, 33 (2008) 2, pp. 439-473