Compactness and structure of zero-states for unoriented Aviles-Giga functionals

TL;DR

This paper extends the Aviles-Giga functional to unoriented fields, introducing new tools like a nonlinear curl operator and trigonometric entropies, leading to a compactness theorem and a detailed characterization of zero-states.

Contribution

It develops an unoriented version of the Aviles-Giga functional with novel mathematical tools, providing new compactness and zero-state characterization results.

Findings

Sequences with bounded energy are precompact.

Zero-states are Lipschitz except at finite points.

Zero-states exhibit vortex or disclination patterns.

Abstract

Motivated by some models of pattern formation involving an unoriented director field in the plane, we study a family of unoriented counterparts to the Aviles-Giga functional. We introduce a nonlinear curl operator for such unoriented vector fields as well as a family of even entropies which we call "trigonometric entropies". Using these tools we show two main theorems which parallel some results in the literature on the classical Aviles-Giga energy. The first is a compactness result for sequences of configurations with uniformly bounded energies. The second is a complete characterization of zero-states, that is, the limit configurations when the energies go to 0. These are Lipschitz continuous away from a locally finite set of points, near which they form either a vortex pattern or a disclination with degree 1/2. The proof is based on a combination of regularity theory together with techniques coming from the study of the Ginzburg-Landau energy. Our methods provide alternative proofs in the classical Aviles-Giga context.