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

We establish a unique continuation property for solutions of the differential inequality $\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }\le V\mathrm{\mid }u\mathrm{\mid }$, 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 $\bar{\mathrm{\partial }}$ and characterize the vanishing order for smooth extension of holomorphic functions across the boundary.