Calculus of Variations and Geometric Measure Theory

Variational methods and applications

The unoriented Aviles-Giga functional for pattern formation

Benoit Merlet (Université de Lille)

created by paolini on 08 Sep 2021
modified on 14 May 2022

9 sep 2021 -- 16:00   [open in google calendar]


We propose a non-oriented version of the Aviles--Giga functional. Recall that it is defined for curl-free vector fields $u_\varepsilon\in H^1(\Omega\subset\mathbf{R}^2,\mathbf{R}^2)$ by \[ \dfrac{\varepsilon}2 \int_\Omega
\nabla u_\varepsilon
^2 \,+\dfrac1{2\varepsilon}\int_\Omega (1-
^2)^2\,,\] where $\varepsilon>0$ is a small parameter. We modify this functional by identifying opposite points $\pm u$ in the target space.\\ The motivation stems from models of stripe pattern formations in the plane. The main purpose of the unoriented version of Aviles--Giga fuctional is to allow for disclinations of degree $1/2$ with a negligible cost as $\varepsilon$ goes to 0.\\ We prove a compactness result for families $\{u_\varepsilon\}_{\varepsilon\downarrow0}$ with uniformely bounded energies and we study the $\Gamma$-limit of these energies : we establish upper bounds by essentially explicit constructions and lower bounds by calibration. The most surprising result is that, unlike in the oriented case, the curl-free constraint may be lost by passing to the limit: the limit energies are finite on $BV(\Omega,\mathbf{S}^1/\{-u\sim u\})$.

joint work with Michael Goldman, Marc PĂ©gon and Sylvia Serfaty.