Preprint 21/2004

A Two Well Liouville Theorem

Andrew Lorent

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Submission date: 20. Apr. 2004 (revised version: January 2006)
Pages: 48
MSC-Numbers: 74N15
Keywords and phrases: two wells, surface energy, liouville theorem
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Abstract:
In this paper we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller.

Let formula24 be a formula26 diagonal matrix. Let formula28. Let formula30. Let formula32 be a formula34 invertible bilipschitz function with formula36, formula38.

There exists positive constants formula40 and formula42 depending only on formula44, formula46, formula48 such that if formula50 and u satisfies the following inequalities
displaymath54

displaymath56
then there exists formula58 and formula60 such that
displaymath62

03.04.2017, 12:08