Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 1 10.wxyz/CV72P1 http://www.austms.org.au/Publ/Bulletin/V72P1/ Xiangxing Tao Songyan Zhang 14 February 2006 2006 2 14 67 85 721-5042-TaoZh-2005 10.wxyz/C2005V72P1p67 http://www.austms.org.au/Publ/Bulletin/V72P1/721-5042-TaoZh/ Let $u$ be a solution to a second order elliptic equation with singular potentials belonging to the Kato--Fefferman--Phong's class in Lipschitz domains. We prove the boundary unique continuation theorems and the doubling properties for $u^2$ near the boundary under the zero Neumann boundary condition. 35B60, 35R05, 31B25 MR2162295 02212187 The work of the first author is supported by National Nature Science Foundation of China (No.10471069), and Zhejiang Provincial Natural Science Foundation of China (No.102066). The second author is supported by Scientific Research Fund of Zhejiang Provincial Education Department (No.20040962) and Doctoral Foundation of Ningbo City (No.2004A610003). V. Adolfsson and L. Escauriaza Comm. Pure Appl. Math. MR1466583 V. Adolfsson and L. Escauriaza; $C^{1,\alpha }$ domains and unique continuation at the boundary, \textit{Comm. Pure Appl. Math.} \textbf{50} (1997), pp.~935--969. V. Adolfsson, L. Escauriaza and C. Kenig Rev. Mat. Iberoamericana MR1363203 V. Adolfsson, L. Escauriaza and C. Kenig; Convex domains and unique continuation at the boundary, \textit{Rev. Mat. Iberoamericana} \textbf{11} (1995), pp.~513-1525. M. Aizenman and B. Simon Comm. pure Appl. Math. MR644024 M. Aizenman and B. Simon; Brownian motion and Harnack's inequality for Schr\"{o}dinger operators, \textit{Comm. pure Appl. Math.} \textbf{35} (1982), pp.~209--273. F. Chiarenza, E. B. Fabes and N. Garofalo Proc. Amer. Soc. MR857933 F. Chiarenza, E. B. Fabes and N. Garofalo; Harnack's inequality for Schr\"{o}dinger operators and the continuity of solutions, \textit{Proc. Amer. Soc.} \textbf{98} (1986), pp.~415--425. E. Fabes, N. Garofalo and F-H. Lin J. Funct. Anal. MR1033920 E. Fabes, N. Garofalo and F-H. Lin; A partial answer to a conjecture of B. Simon concerning unique continuation, \textit{J. Funct. Anal.} \textbf{88} (1990), pp.~194--210. N. Garofalo and F-H. Lin Indiana Univ. Math. J. MR833393 N. Garofalo and F-H. Lin; Monotonicity properties of variational integrals, $Ap$ weights and unique continuation, \textit{Indiana Univ. Math. J.} \textbf{35} (1986), pp.~245--268. N. Garofalo and F-H. Lin Comm. Pure Appl. Math. MR882069 N. Garofalo and F-H. Lin; Unique continuation for elliptic operators: a geometric- variational approach, \textit{Comm. Pure Appl. Math.} \textbf{40} (1987), pp.~347--366. C.E. Kenig CBMS Regional Conference Series in Mathematics 83 Amercan Mathematical Society MR1282720 C.E. Kenig; \textit{Harmonic analysis techniques for second order elliptic boundary value problems}, CBMS Regional Conference Series in Mathematics 83 (Amercan Mathematical Society, Providence R.I., 1994). K. Kurata Comm. Partial Differential Equations MR1233189 K. Kurata; A unique continuation theorem for uniformly elliptic equations with strongly singular potentials, \textit{Comm. Partial Differential Equations} \textbf{18} (993), pp.~1161--1189. K. Kurata Proc. Amer. Soc. Math. MR1363173 K. Kurata; A unique continuation theorem for the Schr\"odinger equation with singular magnetic field, \textit{Proc. Amer. Soc. Math.} \textbf{125} (1997), pp.~853--860. B. Simon Bull. Amer. Math. Soc. MR670130 B. Simon; Schr\"odinger semigroups, \textit{Bull. Amer. Math. Soc.} \textbf{7} (1982), pp.~447--521. X.X. Tao Studia Math. MR1891539 X.X. Tao; Doubling properties and unique continuation at the boundary for elliptic operators with singular magnetic fields, \textit{Studia Math.} \textbf{151} (2002), pp.~31--48. G. Verchota J. Funct. Anal. MR769382 G. Verchota; Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, \textit{J. Funct. Anal.} \textbf{59} (1984), pp.~572--611.