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

Michael Goldman (1) , Benoît Merlet (2, 3, 4) , Marc Pegon (2, 3) , Sylvia Serfaty (5)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 RAPSODI - Reliable numerical approximations of dissipative systems
3 LPP - Laboratoire Paul Painlevé - UMR 8524
4 Department of Electronic & Electrical Engineering
5 CIMS - Courant Institute of Mathematical Sciences [New York]

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.

Mots clés: Aviles-Giga functional; entropies; compactness; rigidity; vortices; disclinations; zero-states

Domaines: Equations aux dérivées partielles [math.AP]

Michael Goldman, Benoît Merlet, Marc Pegon, Sylvia Serfaty. Compactness and structure of zero-states for unoriented Aviles-Giga functionals. Journal of the Institute of Mathematics of Jussieu, In press, pp.1-42. ⟨10.1017/S1474748023000075⟩. ⟨hal-03472756⟩