Conference on Calculus of Variations in Lille - 3rd edition - July 4-6 2022

Vortex sheet solutions for the Ginzburg–Landau system in cylinders

Jul 5, 2022, 2:00 PM

1h

M2 building, Cité Scientifique - Meeting room, 1st floor (Laboratoire Paul Painlevé)

Speaker

Radu Ignat (Université Toulouse III - Paul Sabatier )

Description

We consider the Ginzburg-Landau energy $E_{ϵ}$ for ${\mathbb{R}}^M$-valued maps defined in a cylinder $B^N\times (0,1{)}^n$ satisfying the degree-one vortex boundary condition on $\partial B^N\times (0,1{)}^n$ in dimensions $M\ge N\ge 2$ and $n\ge 1$. The aim is to study the radial symmetry of global minimizers of this variational problem. We prove the following: if $N\ge 7$, then for every $ϵ>0$, there exists a unique global minimizer which is given by the non-escaping radially symmetric vortex sheet solution $u_{ϵ}(x,z)=(f_{ϵ}(|x|)\frac{x}{|x|},0_{{\mathbb{R}}^{M-N}})$, $\mathrm{\forall }x\in B^N$ that is invariant in $z\in (0,1{)}^n$. If $2\le N\le 6$ and $M\ge N+1$, the following dichotomy occurs between escaping and non-escaping solutions: there exists ${ϵ}_N>0$ such that

$\bullet$ if $ϵ\in (0,{ϵ}_N)$, then every global minimizer is an escaping radially symmetric vortex sheet solution of the form $R{\tilde{u}}_{ϵ}$ where ${\tilde{u}}_{ϵ}(x,z)=({\tilde{f}}_{ϵ}(|x|)\frac{x}{|x|},0_{{\mathbb{R}}^{M-N-1}},g_{ϵ}(|x|))$ is invariant in $z$-direction with $g_{ϵ}>0$ in $(0,1)$ and $R\in O(M)$ is an orthogonal transformation keeping invariant the space ${\mathbb{R}}^N\times \{0_{{\mathbb{R}}^{M-N}}\}$;

$\bullet$ if $ϵ\ge {ϵ}_N$, then the non-escaping radially symmetric vortex sheet solution $u_{ϵ}(x,z)=(f_{ϵ}(|x|)\frac{x}{|x|},0_{{\mathbb{R}}^{M-N}})$, $\mathrm{\forall }x\in B^N,z\in (0,1{)}^n$ is the unique global minimizer; moreover, there are no bounded escaping solutions in this case.

We also discuss the problem of vortex sheet ${\mathbb{S}}^{M-1}$-valued harmonic maps.