Existence of Ginzburg-Landau minimizers with optimal boundary data and applications

TL;DR

This paper advances the analysis of Ginzburg-Landau minimizers by establishing optimal boundary regularity conditions and exploring applications to energy-minimizing frames in domains with Weil-Petersson boundary curves.

Contribution

It introduces optimal regularity assumptions for boundary data and extends Ginzburg-Landau analysis to new geometric settings involving Weil-Petersson curves.

Findings

Established optimal boundary regularity conditions for Ginzburg-Landau minimizers.

Defined natural energy-minimizing frames for domains with Weil-Petersson boundary curves.

Extended classical analysis to broader geometric contexts.

Abstract

We perform the classical Ginzburg-Landau analysis originated from the celebrated paper by Bethuel, Brezis, H\'elein for optimal boundary data. More precisely, we give optimal regularity assumptions on the boundary curve of planar domains and Dirichlet boundary data on them. When the Dirichlet boundary data is the tangent vector field of the boundary curve, our framework allows us to define a natural energy minimizing frame for simply connected domains enclosing Weil-Petersson curves.