Klein-Gordon equation on a compact, two dimensional domain Consider the Klein-Gordon equation in two dimensions on any compact subset of $\mathbb{R}^2$ (that is, a Jordan domain). The equation is hyperbolic, and since the domain is compact it is not evident that the Cauchy problem is well posed. The question is then, is it possible to prove that a solution does(-not) exist, under which conditions, and what kind of initial value problem can we formulate?
Note that this is not a globally hyperbolic manifold (the presence of the boundary prevents it being globally hyperbolic, among several other conditions not met here) so the classical theory of the KG equations is not directly applicable.
 A: Some work along these lines has been done by Bob Wald and his colleagues over the years:


*

*Wald, R.  Dynamics in nonglobally hyperbolic, static space‐times. J. Math. Phys. 21, 2802 (1980).

*Ishibashi, A. & R. Wald.  Dynamics in non-globally-hyperbolic static spacetimes: II. General analysis of prescriptions for dynamics.  Class. Quantum Grav. 20, 3815 (2003).

*Seggev, I. Dynamics in stationary, non-globally hyperbolic spacetimes.  Class. Quantum Grav. 21, 2651 (2004).


In the original paper, Wald considered the case where the domain $\mathcal{M}$ has a timelike hypersurface $\Sigma$ and a timelike Killing vector field $t^a$ with complete orbits that intersect $\Sigma$ orthogonally at exactly one point.  He showed that it is possible to define an initial-value formulation as follows:

*

*Decompose the Klein-Gordon operator into a "spatial part" $A$ and time derivatives using the Killing vector $t^a$.

*Note that $A$ is a symmetric operator on the Hilbert space $\mathcal{H} = L^2(\Sigma)$ (with respect to some measure depending on $t^a$.  It therefore has at least one self-adjoint extension $A_E$ (the Friedrichs extension) and may have more than one.  If there is a freedom to choose among different self-adjoint extensions, this effectively corresponds to different choices of boundary conditions at the "edges" of the domain.

*Since $A_E$ is a self-adjoint operator, we can define the operators $\cos (A_E t)$ and $\sin(A_E t)$, and explicitly write down a solution for the Klein-Gordon equation on $\mathcal{M}$ in terms of initial data on $\Sigma$.

The later Ishibashi & Wald paper showed that this family of prescriptions is in some sense "natural" under certain conditions (including an energy-conservation law);  and the Seggev paper extended some of these results to stationary spacetimes, where Killing vector field is not hypersurface-orthogonal and so the decomposition of the Klein-Gordon operator is not quite so "neat" as in the static case.
Unfortunately, the case you are considering, where $\mathcal{M}$ is a compact subset of 2-D Minkowski space, does not fit entirely neatly into the assumptions above.  In particular, the assumption of geodesic completeness is a "bad fit" for a compact subset of spacetime, since all geodesics will encounter the "future bound" of $\mathcal{M}$ in some finite amount of time.  One could imagine "extending" $\mathcal{M}$ to some larger geodesically complete subset of Minkowski spacetime, doing the analysis there, and then restricting the solutions so obtained to $\mathcal{M}$.  This would definitely yield a solution on $\mathcal{M}$, but I suspect that it would not have the same "naturalness" properties proved by Ishibashi & Wald.
