发表机构
Center for Applied Mathematics, Renmin University of China; Department of Mathematics, University of Manchester（中国人民大学应用数学中心; 曼彻斯特大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究带角域上Ginzburg--Landau临界点的渐近行为，通过加权分裂和重整化能量，证明极限映射的涡旋位置为重整化能量的临界点。
AI 中文摘要
我们研究了二维Ginzburg--Landau能量在带角域Ω上的临界点U_ε，并施加了Dirichlet边界条件，该条件在顶点V处精确为零。我们通过加权的Lassoued--Mironescu分裂分离出受迫的标量边界层ρ_ε。假设约化对数能量界和顶点对数紧性，我们证明在取子序列后，归一化映射v_ε = U_ε/ρ_ε在远离有限内部涡旋集合A的C^k_loc(Ω\A)中收敛（对所有k≥0）。极限映射是一个典型的S^1值调和映射u_*，包含整数内部涡旋因子和固定的分数角因子。通过移除内部涡旋周围的圆盘和顶点周围的扇形，并减去相应的对数发散，定义了重整化能量。极限映射的内部涡旋位置是该重整化能量的临界点。
英文摘要
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.
