2022/03/10 by Xavier Lamy, Lamy, Xavier, Andrew Lorent +3
Computer Science · Mathematics · #35J60 #49K99 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Approximation and Integration #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2203.05418
openalex publication_date 2022/03/10 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
Given any strictly convex norm ‖⋅‖ on ℝ2 that is C1 in ℝ2∖\0\, we study the generalized Aviles-Giga functional Iε(m):=∫Ω (ε|∇ m|2 + \frac1ε(1-‖m‖2)2) dx, for Ω⊂\mathbb R2 and m\colonΩ→\mathbb R2 satisfying ∇⋅ m=0. Using, as in the euclidean case ‖⋅‖=|⋅|, the concept of entropies for the limit equation ‖m‖=1, ∇⋅ m=0, we obtain the following. First, we prove compactness in Lp 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 ‖⋅‖=|⋅|, and the last two points are sensitive to the anisotropy of the norm ‖⋅‖.