  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.
