On solutions of elliptic equations that decay rapidly on paths

Abstract

Let P ( D ) P(D) be an elliptic differential operator on R n {\mathbb {R}^n} with constant coefficients. It is known that if u is a solution of P ( D ) u = 0 P(D)u = 0 on an unbounded domain and if u decays uniformly and sufficiently rapidly, then u = 0 u = 0 . In this note it is shown that the same conclusion holds if u decays rapidly, but not a priori uniformly, on a sufficiently large set of unbounded paths.