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

Abstract

Abstract Given any strictly convex norm on that is C 1 in we study the generalized Aviles-Giga functional for and satisfying Using, as in the euclidean case the concept of entropies for the limit equation 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