On a generalized Aviles-Giga functional: compactness, zero-energy states, regularity estimates and energy bounds (2203.05418v1)

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 $|\cdot|$.