A UNIQUENESS THEOREM FOR HARMONIC-FUNCTIONS

The main result of this note is the following theorem: Theorem 1. Let D = {(x,t); \x\(2) + t(2) less than or equal to r(2), t > 0} be a half ball in Rn+1 and x epsilon R-n. Assume that u is C-1 in (D) over bar and harmonic in D, and that for every positive integer N there exists a constant C-N such that (1) \del u(x,0)\ less than or equal to C-N\x\ (N) in a neighborhood V of the origin in partial derivative D; (2) u(x ,0) greater than or equal to u(0,0) in V. Then u = u(0,0). First we pr ove it for R-2, and then we show by induction that it holds for all n greater than or equal to 3.