Unique continuation for a gradient inequality with Ln potential

Adam Coffman, Yifei Pan, and Yuan Zhang

article abstract

We establish a unique continuation property for solutions of the
differential inequality |nabla u|<= V|u|, where V is locally Ln
integrable on a domain in Rn. A stronger uniqueness result is obtained
if in addition the solutions are locally Lipschitz. One application is
a finite order vanishing property in the L2 sense for the exponential
of W1,n functions. We further discuss related results for the
Cauchy-Riemann operator d-bar and characterize the vanishing order for
smooth extension of holomorphic functions across the boundary.