On a generalized Aviles-Giga functional: compactness, zero-energy states, regularity estimates and energy bounds

TL;DR

This paper extends the analysis of the Aviles-Giga functional to general convex norms, establishing compactness, rigidity, regularity, and energy bounds for divergence-free vector fields with norm constraints.

Contribution

It generalizes key results from the Euclidean case to arbitrary convex norms, providing new compactness, rigidity, and regularity estimates for the generalized functional.

Findings

Proved compactness in $L^p$ for bounded energy sequences.

Established rigidity of zero-energy states.

Derived optimal regularity estimates based on entropy production.

Abstract

Given any strictly convex norm $\|\cdot\|$ on $\mathbb{R}^2$ that is $C^1$ in $\mathbb{R}^2\setminus\{0\}$, we study the generalized Aviles-Giga functional \[I_{\epsilon}(m):=\int_{\Omega} \left(\epsilon \left|\nabla m\right|^2 + \frac{1}{\epsilon}\left(1-\|m\|^2\right)^2\right) \, dx,\] for $\Omega\subset\mathbb R^2$ and $m\colon\Omega\to\mathbb R^2$ satisfying $\nabla\cdot m=0$. Using, as in the euclidean case $\|\cdot\|=|\cdot|$, the concept of entropies for the limit equation $\|m\|=1$, $\nabla\cdot m=0$, we obtain the following. First, we prove compactness in $L^p$ of sequences of bounded energy. Second, we prove rigidity of zero-energy states (limits of sequences of vanishing energy), generalizing and simplifying a result by Bochard and Pegon. Third, we obtain optimal regularity estimates for limits of sequences of bounded energy, in terms of their entropy productions. Fourth, in the case of a limit map in $BV$, we show that lower bound provided by entropy productions and upper bound provided by one-dimensional transition profiles are of the same order. The first two points are analogous to what is known in the euclidean case $\|\cdot\|=|\cdot|$, and the last two points are sensitive to the anisotropy of the norm $\|\cdot\|$.