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

Goldman, Michael; Merlet, Benoît; Pegon, Marc; Serfaty, Sylvia

Document type :

Article dans une revue scientifique: Article original

DOI :

10.1017/S1474748023000075

Title :

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

Author(s) :

Goldman, Michael [Auteur]
Laboratoire Jacques-Louis Lions [LJLL (UMR_7598)]
Merlet, Benoît [Auteur]

Laboratoire Paul Painlevé - UMR 8524 [LPP]
Reliable numerical approximations of dissipative systems [RAPSODI]
Pegon, Marc [Auteur]

Laboratoire Paul Painlevé - UMR 8524 [LPP]
Reliable numerical approximations of dissipative systems [RAPSODI]
Serfaty, Sylvia [Auteur]
Courant Institute of Mathematical Sciences [New York] [CIMS]

Journal title :

Journal of the Institute of Mathematics of Jussieu

Pages :

941-982

Publisher :

Cambridge University Press (CUP)

Publication date :

2024-03-10

ISSN :

1474-7480

English keyword(s) :

Aviles-Giga functional
entropies
compactness
rigidity
vortices
disclinations
zero-states

HAL domain(s) :

Mathématiques [math]/Equations aux dérivées partielles [math.AP]

English abstract : [en]

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 ...
Show more >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.Show less >

Language :

Anglais

Peer reviewed article :

Oui

Audience :

Internationale

Popular science :

Non

ANR Project :

Centre Européen pour les Mathématiques, la Physique et leurs Interactions
Optimisation de forme

Collections :