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

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+1ϵ(1−‖m‖2)2) dx, for Ω⊂R2 and m:Ω→R2 satisfying ∇·m=0. Using, as euclidean case ‖·‖=|·|, concept of entropies limit equation ‖m‖=1, ∇·m=0, obtain following. First, prove compactness Lp sequences bounded energy. Second, rigidity zero-energy states (limits vanishing energy), generalizing simplifying a result by Bochard Pegon. Third, optimal regularity estimates limits energy, terms their entropy productions. Fourth, map BV, show lower bound provided productions upper one-dimensional transition profiles are same order. The first two points analogous to what known last sensitive anisotropy ‖·‖.