The Asymptotic Theory of Ginzburg--Landau Critical Points on Domains with Corners

Abstract

We study the critical points $U_\varepsilon$ of the two-dimensional Ginzburg--Landau energy on domains with corners $Ω$ and imposed Dirichlet boundary condition which vanishes precisely at the vertices $V$. We isolate the forced scalar boundary layer $ρ_\varepsilon$ by the weighted Lassoued--Mironescu splitting. Assuming the reduced logarithmic energy bound and vertex logarithmic tightness, we prove that after passing to a subsequence, the normalized maps $v_\varepsilon = U_\varepsilon/ρ_\varepsilon$ converges in $C^k_{\mathrm{loc}}(Ω\setminus A)$ $\forall k\ge0$ away from a finite set of interior vortices $A$. The limiting map is a canonical $S^1$-valued harmonic map $u_*$ containing integer interior vortex factors and fixed fractional corner factors. The renormalized energy is defined by removing discs around the interior vortices and sectors around the vertices and subtracting the corresponding logarithmic divergences. The interior vortex locations of the limiting map are critical points of this renormalized energy.