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COMPACTNESS AND STRUCTURE OF ZERO-STATES FOR UNORIENTED AVILES–GIGA FUNCTIONALS

Part of: Hyperbolic equations and systems Partial differential equations Manifolds Elliptic equations and systems Equilibrium (steady-state) problems Variational principles of physics

Published online by Cambridge University Press:  10 March 2023

M. Goldman ,
B. Merlet Open the ORCID record for B. Merlet [Opens in a new window] ,
M. Pegon  and
S. Serfaty
M. Goldman
Affiliation:
Université de Paris and Sorbonne Université, CNRS, LJLL, F-75005 Paris, France (michael.goldman@u-paris.fr)
B. Merlet*
Affiliation:
Univ. Lille, CNRS, UMR 8524, Inria - Laboratoire Paul Painlevé, F-59000 Lille
M. Pegon
Affiliation:
Univ. Lille, CNRS, UMR 8524, Inria - Laboratoire Paul Painlevé, F-59000 Lille (marc.pegon@univ-lille.fr)
S. Serfaty
Affiliation:
Courant Institute of Mathematical Sciences, 251 Mercer St., New York NY 10012, USA (serfaty@cims.nyu.edu)
*
benoit.merlet@univ-lille.fr
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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 $\operatorname {\mathrm {curl}}$ 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.

MSC classification

Primary: 49Q20: Variational problems in a geometric measure-theoretic setting
Secondary: 35B36: Pattern formation 35B65: Smoothness and regularity of solutions 35J60: Nonlinear elliptic equations 35L65: Conservation laws 49S05: Variational principles of physics (should also be assigned at least one other classification number in section 49) 74G65: Energy minimization
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 23 , Issue 2 , March 2024 , pp. 941 - 982
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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