Calculus of Variations and Geometric Measure Theory

A. Lorent

A quantitative characterisation of functions with low Aviles Giga energy on convex domains

created by lorent on 01 Feb 2009
modified on 18 Feb 2014


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


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