I'm working on problems related to the inhomogeneous Klein–Gordon equation ($[\partial_t^2 - \nabla^2 + m^2]\phi(t,\mathbf{x}) = F(t,\mathbf{x})$), and I started to consider how general I could make the boundary conditions. For elliptic differential equations I know that you can have Dirichlet ($\phi$ fixed on boundary), Neumann ($\partial_{\hat{n}}\phi \equiv \hat{n}\cdot\nabla\phi$ fixed on boundary), and Robin (linear combination of Dirichlet & Neumann) boundary conditions. The Robin boundary conditions can be generalized in a fairly straightforward way by considering a local function of $\phi$ and $\partial_{\hat{n}}\phi$.
In detail, if we assume that \begin{align} f(\phi, \partial_{\hat{n}}\phi,\mathbf{x}) &= \alpha(\mathbf{x}) \end{align} is the boundary condition then, as long as $f$ satisfies \begin{align} \frac{\partial^2 f}{\partial \phi^2} + \frac{\partial^2 f}{\partial (\partial_{\hat{n}}\phi)^2} &= 0 \\ \left(\frac{\partial f}{\partial \phi}\right)^2 + \left(\frac{\partial f}{\partial (\partial_{\hat{n}}\phi)}\right)^2 & > 0 \end{align} for all $\mathbf{x}$, then we can construct a harmonic conjugate to $f$ called $g$. Because the relationship between $(\phi,\partial_{\hat{n}}\phi)$ and $(f,g)$ is everywhere invertible, we can then use $g$ in the process of working out what boundary conditions the Green's function has to satisfy to construct a solution for $\phi$ that satisfies the static Klein–Gordon equation (generalization of Poisson's equation) and boundary conditions.
With the full Klein–Gordon equation we have the addition of Cauchy boundary conditions (initial/final value problems). Is there a similarly general way to construct the boundary conditions in this problem? I mean, we could consider the generalization of Robin-like boundary conditions for the initial and final space-like time slices in the exact same way, but I'm having trouble thinking of a similarly general mixing in of Cauchy type boundary conditions. The sticking point is that the Robin-type mixing of Dirichlet and Neumann is purely local. With Cauchy boundary conditions you over-specify data on one boundary and leave the data on the other boundary completely unspecified. This is a non-local interchange, and because of this I cannot think of how to interpolate among initial, final, and Robin boundary conditions in a way that is assured to keep the problem well-posed.