The Aviles-Giga functional I ǫ (u) = Ω |1−|∇u| 2 | 2 ǫ + ǫ ∇ 2 u 2 dx is a well known second order functional that models phenomena from blistering to liquid crystals. The zero energy states of the Aviles-Giga functional have been characterized by Jabin, Otto, Perthame [Ja-Ot-Pe 02]. Among other results they showed that if lim n→∞ I ǫn (u n ) = 0 for some sequence u n ∈ W 2,2 0 (Ω) and u = lim n→∞ u n then ∇u is Lipschitz continuous outside a finite set. This is essentially a corollary to their theorem that if u is a solution to the Eikonal equation |∇u| = 1 a.e. and if for every "entropy" Φ (in the sense of [De-Mu-Ko-Ot 01], Definition 1) function u satisfies ∇ · Φ(∇u ⊥ ) = 0 distributionally in Ω then ∇u is locally Lipschitz continuous outside a locally finite set.In this paper we generalize this result by showing that if u satisfies the Eikonal equation and ifwhere Σ e 1 e 2 and Σ ǫ 1 ǫ 2 are the entropies introduced by Ambrosio, DeLellis, Mantegazza [Am-De-Ma 99], Jin, Kohn [Ji-Ko 00], then ∇u is locally Lipschitz continuous outside a locally finite set. Condition (1) being fairly natural this result could also be considered a contribution to the study of the regularity of solutions of the Eikonal equation. The method of proof is to transform any solution of the Eikonal equation satisfying (1) into a differential inclusion DF ∈ K where K ⊂ M 2×2 is a connected compact set of matrices without Rank-1 connections. Equivalently this differential inclusion can be written as a constrained non-linear Beltrami equation. The set K is also non-elliptic in the sense of Sverak [Se 93]. By use of this transformation and by utilizing ideas from the work on regularity of solutions of the Eikonal equation in fractional Sobolev space by Ignat [Ig 12], DeLellis, Ignat [De-Ig 15] as well as methods of Sverak [Se 93], regularity is established. 2000 Mathematics Subject Classification. 28A75.

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We analyze the Lawrence-Doniach model for three-dimensional highly anisotropic superconductors with layered structure. For such a superconductor occupying a bounded generalized cylinder in R 3 with equally spaced parallel layers, we assume an applied magnetic field that is perpendicular to the layers with intensity hex ∼ | ln ǫ| as ǫ → 0, where ǫ is the reciprocal of the Ginzburg-Landau parameter. We prove Gamma-convergence of the Lawrence-Doniach energy as ǫ and the interlayer distance s tend to zero, under the additional assumption that the layers are weakly coupled (i.e., s ≫ ǫ).

We analyze minimizers of the Lawrence-Doniach energy for layered superconductors with Josephson constant λ and Ginzburg-Landau parameter 1/ǫ in a bounded generalizedwhere Ω is a bounded simply connected Lipschitz domain in R 2 . Our main result is that in an applied magnetic field H ex = h ex e 3 which is perpendicular to the layers with |ln ǫ| ≪ h ex ≪ ǫ −2 , the minimum Lawrence-Doniach energy is given by (1)) as ǫ and the interlayer distance s tend to zero. We also prove estimates on the behavior of the order parameters, induced magnetic field, and vorticity in this regime. Finally, we observe that as a consequence of our results, the same asymptotic formula holds for the minimum anisotropic three-dimensional Ginzburg-Landau energy in D with anisotropic parameter λ and o ǫ,s (1) replaced by o ǫ (1).

Given any strictly convex norm • on R 2 that is C 1 in R 2 \ {0}, we study the generalized Aviles-Giga functionalfor 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 • .