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

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.

Vector Field Models for Nematic Disclinations

“…We can also make a connection to the recent preprint [GMPS21], where a SBV model for ± 1 2 disclinations is proposed and the constraint that [k] = 2 along the jump set is imposed. The energy we use can be viewed as an attempt to also relax the one used in [GMPS21] by being a Sobolev model allowing for a more general class of jumps in the SBV limit.…”