Quantitative unique continuation for Neumann problem in planar $C^{1,\alpha }$ domains

In this paper, we study the quantitative unique continuation property of the second-order elliptic operators under the vanishing Neumann boundary condition over $C^{1,\alpha }$ or convex domains in two dimensions. We establish the optimal estimates of the number of critical points, doubling index and the total length of level curves. The key idea is to reduce the Neumann problem to the Dirichlet problem, which has been understood better, by a classical duality between an $A$-harmonic function and its stream function.