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 u 2 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 . p weight properties for the solution u near the boundary.