Mathematics > Analysis of PDEs
[Submitted on 1 Feb 2009 (v1), last revised 23 Feb 2012 (this version, v3)]
Title:A quantitative characterisation of functions with low Aviles Giga energy on convex domains
View PDFAbstract:Given a connected Lipschitz domain U we let L(U) be the subset of functions in 2nd order Sobolev space whose gradient (in the sense of trace) is equal to the inward pointing unit normal to U. The the Aviles Giga functional over L(U) serves as a model in connection with problems in liquid crystals and thin film blisters, it is also the most natural higher order generalisation of the Modica Mortola functional. Jabin, Otto, Perthame characterised a class of functions which includes all limits of sequences whose Aviles Giga energy goes to zero. A corollary to their work is that if there exists such a sequence for a bounded domain U, then U must be a ball and the limiting function must be the distance from the boundary. We prove a quantitative generalisation of this corollary for the class of bounded convex sets.
As a consequence of this we show that if U has C^2 boundary and is close to a ball, then for all small enough \ep the minimiser of I_{\ep} is close to the distance function from the boundary.
Submission history
From: Andrew Lorent [view email][v1] Sun, 1 Feb 2009 19:22:42 UTC (100 KB)
[v2] Mon, 29 Mar 2010 00:44:23 UTC (101 KB)
[v3] Thu, 23 Feb 2012 17:06:53 UTC (108 KB)
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