Asymptotic minimality of one-dimensional transition profiles in Aviles-Giga type models: an approach via 1-currents

TL;DR

This paper investigates the asymptotic behavior of certain energy functionals related to phase transitions in vector fields, revealing conditions under which transition profiles become effectively one-dimensional using geometric measure theory tools.

Contribution

It introduces a new geometric variational problem involving 1-currents to analyze energy concentration and transition profiles in Aviles-Giga type models.

Findings

Energy concentration estimates via 1-currents

Conditions for one-dimensional transition profiles

Asymptotic analysis of Modica-Mortola functionals

Abstract

For vector fields on a two-dimensional domain, we study the asymptotic behaviour of Modica-Mortola (or Allen-Cahn) type functionals under the assumption that the divergence converges to $0$ at a certain rate, which effectively produces a model of Aviles-Giga type. This problem will typically give rise to transition layers, which degenerate into discontinuities in the limit. We analyse the energy concentration at these discontinuities and the corresponding transition profiles. We derive an estimate for the energy concentration in terms of a novel geometric variational problem involving the notion of $\mathbb{R}^2$-valued $1$-currents from geometric measure theory. This in turn leads to criteria, under which the energetically favourable transition profiles are essentially one-dimensional.