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

Abstract

On a compact manifold $M^n$ ($n\ge 3$) 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 $\partial M$.