On the Unique Continuation Properties for Elliptic Operators with Singular Potentials

Let u be a solution to a second order elliptic equation with singular potentials belonging to Kato–Fefferman–Phong’s class in Lipschitz domains. An elementary proof of the doubling property for u2 over balls is presented, if the balls are contained in the domain or centered at some points near an open subset of the boundary on which the solution u vanishes continuously. Moreover, we prove the inner unique continuation theorems and the boundary unique continuation theorems for the elliptic equations, and we derive the ℬp weight properties for the solution u near the boundary.