Unique continuation for a gradient inequality with $L^n$ potential

Abstract

We establish a unique continuation property for solutions of the differential inequality $|\nabla u|\le V|u|$, where $V$ is locally $L^n$ integrable on a domain in ${\mathbb{R}}^n$. A stronger uniqueness result is obtained if in addition the solutions are locally Lipschitz. One application is a finite order vanishing property in the $L^2$ sense for the exponential of $W^{1,n}$ functions. We further discuss related results for the Cauchy-Riemann operator $\bar{\partial }$ and characterize the vanishing order for smooth extension of holomorphic functions across the boundary.

Version 1 supersedes and significantly expands an earlier arxiv submission by the second author, arXiv:2210.04364. Version 2 includes minor corrections and edits after referee report