Let us consider the Klein-Gordon equation $$(\Box +m^2)u=0,$$ where $u$ is a scalar valued function, $m\geq 0$, $\Box=\frac{\partial^2}{\partial x_0^2}-\sum_{i=1}^d\frac{\partial^2}{\partial x_i^2}$.

In the physics literature, in particular in QFT, one sometimes says that if the unknown function $u$ satisfies boundary conditions then $u=0$.

Question. What exactly boundary conditions guarantee vanishing of $u$?

I am mainly interested in such conditions of physical interest, say used in QFT.