*Accepted Paper*

**Inserted:** 1 feb 2009

**Last Updated:** 18 feb 2014

**Journal:** Ann. Sc. Norm. Super. Pisa. Cl. Sci.

**Volume:** XIII

**Number:** 5

**Pages:** 1-66

**Year:** 2014

**Abstract:**

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.

**Keywords:**
Aviles Giga functional

**Download:**