The unoriented Aviles-Giga functional for pattern formation
speaker: Benoit Merlet (Université de Lille)abstract:
We propose a non-oriented version of the Aviles--Giga functional. Recall that it is defined for curl-free vector fields $u\varepsilon\in H1(\Omega\subset\mathbf{R}2,\mathbf{R}2)$ by \[ \dfrac{\varepsilon}2 \int_\Omega
\nabla u_\varepsilon
^2 \,+\dfrac1{2\varepsilon}\int_\Omega (1-
u_\varepsilon
^2)^2\,,\] 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 $12$ 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 Pégon and Sylvia Serfaty.
timetable:
Thu 9 Sep, 16:00 - 16:50, Aula Dini
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