Unique continuation for a gradient inequality with Ln potential

Abstract

We establish a unique continuation property for solutions of the differential inequality $\left|\nabla u\right|\le V\left|u\right|$, where V is locally $L^n$ integrable on a domain in $R^n$. 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 $\overline{\partial }$ and characterize the vanishing order for smooth extension of holomorphic functions across the boundary.