Asymptotics for the Ginzburg-Landau equation on manifolds with boundary under homogeneous Neumann condition

TL;DR

This paper investigates the asymptotic behavior of solutions to the Ginzburg-Landau equation on manifolds with boundary, revealing energy concentration phenomena and the emergence of harmonic forms and varifolds under Neumann boundary conditions.

Contribution

It provides a detailed analysis of the energy decomposition and concentration phenomena for Ginzburg-Landau solutions on manifolds with boundary, under Neumann conditions, extending understanding of vortex structures.

Findings

Energy splits into harmonic form and varifold concentration.

Varifold can be supported on boundary or vanish.

Results depend on boundary convexity and energy bounds.

Abstract

On a compact manifold $M^{n}$ ($n\geq 3$) with boundary, we study the asymptotic behavior as $\epsilon$ tends to zero of solutions $u_{\epsilon}: M \to \mathbb{C}$ to the equation $\Delta u_{\epsilon} + \epsilon^{-2}(1 - |u_{\epsilon}|^{2})u_{\epsilon} = 0$ with the boundary condition $\partial_{\nu}u_{\epsilon} = 0$ on $\partial M$. Assuming an energy upper bound on the solutions and a convexity condition on $\partial M$, we show that along a subsequence, the energy of $\{u_{\epsilon}\}$ breaks into two parts: one captured by a harmonic $1$-form $\psi$ on $M$, and the other concentrating on the support of a rectifiable $(n-2)$-varifold $V$ which is stationary with respect to deformations preserving $\partial M$. Examples are given which shows that $V$ could vanish altogether, or be non-zero but supported only on $\partial M$.