Preprint

Inserted: 10 mar 2022
Last Updated: 10 mar 2022

Year: 2022

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 \[I_{\epsilon}(m):=\int_{\Omega} \left(\epsilon |\nabla m |^2 + \frac{1}{\epsilon}\left(1-||m||^2\right)^2\right) \, dx,\] 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 $||\cdot||$.

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• generalizedavilesgiga.pdf