On solutions of elliptic equations that decay rapidly on paths

Abstract:

Let $P(D)$ be an elliptic differential operator on ${\mathbb {R}^n}$ with constant coefficients. It is known that if u is a solution of $P(D)u = 0$ on an unbounded domain and if u decays uniformly and sufficiently rapidly, then $u = 0$. 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.