On Landis' conjecture in the plane

Abstract

In this paper we prove a quantitative form of Landis' conjecture in the plane. Precisely, let $W(z)$ be a measurable real vector-valued function and $V(z)\ge 0$ be a real measurable scalar function, satisfying $\Vert W{\Vert }_{L^{\infty }({\mathbf{R}}^2)}\le 1$ and $\Vert V{\Vert }_{L^{\infty }({\mathbf{R}}^2)}\le 1$. Let $u$ be a real solution of $\Delta u-\nabla (Wu)-Vu=0$ in ${\mathbf{R}}^2$. Assume that $u(0)=1$ and $|u(z)|\le \exp (C_0|z|)$. Then $u$ satisfies $\underset{|z_0|=R}{\inf }\underset{|z-z_0|<1}{\sup }|u(z)|\ge \exp (-CR\log R)$, where $C$ depends on $C_0$. In addition to the case of the whole plane, we also establish a quantitative form of Landis' conjecture defined in an exterior domain.

correct some typos

References in corpus (2)

• Strong unique continuation for general elliptic equations in 2D
• Some quantitative unique continuation results for eigenfunctions of the magnetic Schrödinger operator

Cited by in corpus (1)

• Quantitative uniqueness estimates for second order elliptic equations with unbounded drift