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arXiv · 2112.04959

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

Michael Goldman·Benoit Merlet·Marc Pegon·Sylvia Serfaty
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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.

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Michael Goldman, Benoit Merlet, Marc Pegon, Sylvia Serfaty. 2021-12-09. Compactness and structure of zero-states for unoriented Aviles-Giga functionals. https://arxiv.org/abs/2112.04959

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