What classical conditions give unique Laplace equation solutions on a half-plane?

Question

Solutions of the Laplace equation on the upper half plane of $\mathbb{R}^{2}$ are not unique: $$ \frac{\partial^{2}}{\partial x^{2}}u+\frac{\partial^{2}}{\partial y^{2}}u = 0,\;\;\; -\infty < x < \infty,\;\; y > 0,\\ u(x,0) = g(x) $$ What types of classical conditions on $g$ and $u$ guarantee unique solutions? I'm interested in any and all types of common, classical conditions that people have seen.