On a generalized Aviles-Giga functional: compactness, zero-energy states, regularity estimates and energy bounds
Résumé
Given any strictly convex norm • on R 2 that is C 1 in R 2 \ {0}, we study the generalized Aviles-Giga functional I (m) := ˆΩ |∇m| 2 + 1 1 − m 2 2 dx, for Ω ⊂ R 2 and m : Ω → R 2 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 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 • = | • |, and the last two points are sensitive to the anisotropy of the norm • .
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