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On a compact manifold $M^{n}$ ($n\geq 3$) with boundary, we study theasymptotic 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 upperbound on the solutions and a convexity condition on $\partial M$, we show thatalong a subsequence, the energy of $\{u_{\epsilon}\}$ breaks into two parts:one captured by a harmonic $1$-form $\psi$ on $M$, and the other concentratingon the support of a rectifiable $(n-2)$-varifold $V$ which is stationary withrespect to deformations preserving $\partial M$. Examples are given which showsthat $V$ could vanish altogether, or be non-zero but supported only on$\partial M$.

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