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

Abstract

Given any strictly convex norm $\Vert ·\Vert$ on ${\mathbb{R}}^2$ that is C¹ in ${\mathbb{R}}^2\setminus \left\{0\right\},$ we study the generalized Aviles-Giga functional $I_{ϵ}(m):={\int }_{\Omega }(ϵ{\left|\nabla m\right|}^2+\frac{1}{ϵ}{(1-\Vert m{\Vert }^2)}^2) dx,$ for $\Omega \subset {\mathbb{R}}^2$ and $m:\Omega \to {\mathbb{R}}^2$ satisfying $\nabla ·m=0.$ Using, as in the euclidean case $\Vert ·\Vert =|·|,$ the concept of entropies for the limit equation $\Vert m\Vert =1,\nabla ·m=0,$ 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 $\Vert ·\Vert =|·|,$ and the last two points are sensitive to the anisotropy of the norm $\Vert ·\Vert .$

Keywords:

• line-energy
• Aviles-Giga
• kinetic formulation

Notes

1 Note in order to get the pointwise convergence in (20(20) $${\Phi }_{\delta }^{\xi }(z)\to {\Phi }^{\xi }(z)\mathrm{ }\mathit{\text{for}}\mathrm{ }\mathit{\text{all}}\mathrm{ }z\in \partial \mathsf{B}.$$(20) ) for every $z\in \partial \mathsf{B}$ we can not define ${\widehat{\lambda }}_{\delta }$ via the standard symmetric (across zero) kernel centered on θ₀, ${\theta }_0+\pi .$ This is why we do the two step procedure of defining ${\widehat{\lambda }}_{\delta }$ then modifying it.

Additional information

Funding

X.L. received support from ANR project ANR-18-CE40-0023. A.L. gratefully acknowledges the support of the Simons foundation, collaboration grant #426900.