Singular Limits for Three-Dimensional Global Minimizers of Ginzburg--Landau-Type Functionals: Uniform Estimates and Singular Sets

Abstract

We study the asymptotic behavior of global minimizers of a Ginzburg--Landau-type functional with general compact vacuum manifold $\mathcal{N}$ on bounded domains in ${\mathbb{R}}^3$, in the regime where the energy grows at a logarithmic rate. We show that the normalized energy measures converge, up to a subsequence, to a measure supported on a finite union of closed line segments connecting prescribed singularities on the boundary. The limit map is a harmonic map valued locally by minimizing $\mathcal{N}$ away from this singular set. We also establish uniform $W^{1,q}$-estimates with $q\in (1,2)$ and uniform potential estimates for minimizers, independent of the parameter $\epsilon$. Finally, we prove that the singular set of the limiting measure solves the homotopical Plateau problem in codimension $2$.

72 pages, 1 figures, comments are welcome!