On Landis' conjecture in the plane

TL;DR

This paper proves a quantitative version of Landis' conjecture for solutions to a certain elliptic PDE in the plane, establishing lower bounds on the solution's growth under specific boundedness conditions.

Contribution

It provides the first quantitative bounds for Landis' conjecture in the plane, including exterior domain cases, for solutions of elliptic equations with bounded measurable coefficients.

Findings

Established a lower bound of rac{-CR\, R ext{log} R} for solutions in or R  the plane.

Extended the quantitative Landis' conjecture to exterior domains.

Demonstrated bounds depend explicitly on the growth parameter C_0.

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 $\|W\|_{L^{\infty}({\mathbf R}^2)}\le 1$ and $\|V\|_{L^{\infty}({\mathbf R}^2)}\le 1$. 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.