Asymptotic minimality of one-dimensional transition profiles in Aviles-Giga type models: an approach via 1-currents
Radu Ignat, Roger Moser
TL;DR
The paper advances the understanding of energy concentration in Aviles–Giga type models by recasting the transition energy into a geometric variational problem over R^2-valued 1-currents. It develops a calibration framework and a regularised F-mass that connect PDE-based transition profiles to a mass minimisation problem, yielding sharp lower bounds and identifying when one-dimensional transition profiles are energetically optimal. Through L∞-minimisation, current decompositions, and Korn-type estimates, the authors provide conditions under which E(a^−,a^+) equals the line-integral of sqrt{W} along the segment [a^−,a^+], and they illustrate the results with Aviles–Giga–type potentials and micromagnetic-inspired microstructures. The approach offers a robust toolkit for determining when one-dimensional profiles prevail and for analyzing complex microstructures via current-based geometric measures, with explicit examples and corollaries.
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.