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

On a compact manifold ($) over bar (n) (n >= 3) with boundary, we study the asymptotic behavior as e tends to zero of solutions u(epsilon) : M -> C to the equation Delta u(is an element of) + epsilon(-2)(1 - vertical bar mu(epsilon)vertical bar(2))mu(epsilon) = 0 with epsilon the boundary condition partial derivative(nu)mu(epsilon) = 0 on partial derivative M. Assuming an energy upper bound on the solutions and a convexity condition on partial derivative 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 derivative M. Examples are given which shows that V could vanish altogether, or be non-zero but supported only on partial derivative M. (C) 2019 Elsevier Inc. All rights reserved.