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) ≥ 0 be a real measurable scalar function, satisfyingC 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.where W (x, y) = (W 1 (x, y), W 2 (x, y)) and V (x, y) are measurable, real-valued, and V (x, y) ≥ 0 a.e. In view of the scaling argument in [BK05], Landis' conjecture is an easy consequence *

On the fractional Landis conjecture

“…Hence, this strongly suggests that for the class of potentials under consideration, Theorem 1 is essentially sharp (possibly up to logarithmic contributions). However, we further believe that as in [KSW15], at least under sign conditions on the potential and in one dimension (which on the level of the Cafferelli-Silvestre extension corresponds to the twodimensional setting from [KSW15] in which complex analysis tools are available), it might be possible to reduce the necessary regularity for q to L ∞ regularity.…”

“…scalar equations), we expect that the exponential decay (independent of the value of s ∈ (0, 1)) as the critical decay behaviour is sharp. Indeed, as in [KSW15] and [BK05] it is possible to relate the decay behaviour at infinity to the local maximal vanishing rate at zero (if growth conditions are assumed, which are necessary due to the global character of the problem). Analogous arguments as in the classical case s = 1 would lead to the conjecture that when considering…”

On Landis’ conjecture in the plane for some equations with sign-changing potentials

“…In recent years, considerable progress has been made towards resolving Landis' conjecture in the real-valued planar setting. In their breakthrough article [KSW15], Kenig, Silvestre, and Wang introduced a new method based on tools from complex analysis to reduce the value of β in (1.1) from 4/3 down to 1. Using the scaling argument first introduced in [BK05], the Landis-type theorems in [KSW15] are consequences of order of vanishing estimates for solutions to local versions of the equation.…”

Quantitative uniqueness estimates for second order elliptic equations with unbounded drift

“…In a recent paper [KLW14], the authors studied Landis' conjecture for second order elliptic equations in the plane in the real setting, including (1.1) with real-valued W and u. It was proved in [KLW14] that if u is a real-valued solution of (1. where C depends on C 0 .…”

“…It was proved in [KLW14] that if u is a real-valued solution of (1. where C depends on C 0 . In this paper, we would like to study estimates like (1.2) for 2 ≤ p < ∞.…”