A UNIQUE CONTINUATION PROPERTY ON THE BOUNDARY FOR SOLUTIONS OF ELLIPTIC-EQUATIONS

We prove the following conclusion: if u is a harmonic function on a smooth domain OMEGA in R(n), n greater-than-or-equal-to 3, or a solution of a general second-order linear elliptic equation on a domain OMEGA in R2, and if there are x0 is-an-element-of partial derivative OMEGA and constants a, b > 0 such that \u(x)\ less-than-or-equal-to a exp {-b/\x-x0\} for x is-an-element-of OMEGA, \x-x0\ small, then u = 0 in OMEGA. The decay rate in our results is best possible by the example that u = real part of exp{-1/z(alpha)}, 0 < alpha < 1, is harmonic but not identically zero in the right complex half-plane.