A unique continuation property on the boundary for solutions of elliptic equations

Abstract:

We prove the following conclusion: if $u$ is a harmonic function on a smooth domain $\Omega$ in ${R^n}$ , $n \geq 3$ , or a solution of a general second-order linear elliptic equation on a domain $\Omega$ in ${R^2}$, and if there are ${x_0} \in \partial \Omega$ and constants $a$, $b > 0$ such that $|u(x)| \leq a\exp \{ - b/|x - {x_0}|\}$ for $x \in \Omega$, $|x - {x_0}|$ 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.