Improved quantitative unique continuation for complex-valued drift equations in the plane

Blair Davey; Carlos Kenig; Jenn-Nan Wang

paper

Improved quantitative unique continuation for complex-valued drift equations in the plane

arXiv:2004.00157

Abstract

In this article, we investigate the quantitative unique continuation properties of complex-valued solutions to drift equations in the plane. We consider equations of the form $Δ u + W \cdot \nabla u = 0$ in $R^{2}$ , where $W = W_{1} + i W_{2}$ with each $W_{j}$ real-valued. Under the assumptions that $W_{j} \in L^{q_{j}}$ for some $q_{1} \in [2, \infty]$ , $q_{2} \in (2, \infty]$ , and $W_{2}$ exhibits rapid decay at infinity, we prove new global unique continuation estimates. This improvement is accomplished by reducing our equations to vector-valued Beltrami systems. Our results rely on a novel order of vanishing estimate combined with a finite iteration scheme.