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

TL;DR

This paper investigates the quantitative unique continuation for second-order elliptic operators with Neumann boundary conditions in planar $C^{1,eta}$ domains, providing optimal estimates for critical points, doubling index, and level curve length.

Contribution

It introduces a novel reduction of the Neumann problem to the Dirichlet problem using duality, leading to new optimal estimates in two-dimensional domains.

Findings

Established optimal bounds on the number of critical points.

Derived estimates for the doubling index.

Quantified the total length of level curves.

Abstract

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.