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

TL;DR

This paper provides a quantitative analysis of functions with low Aviles Giga energy on convex domains, showing that near-minimizers are close to the distance function from the boundary, especially for domains close to a ball.

Contribution

It extends previous characterizations of low-energy functions to a quantitative setting for convex domains, linking domain shape to energy minimizers.

Findings

Minimizers of the Aviles Giga functional are close to the distance function on convex domains.

If the domain is close to a ball, the energy minimizers approximate the distance function.

The results hold for domains with C^2 boundary that are near spherical in shape.

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. 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.