Consider the equation


$$
\begin{equation}
\frac{\partial^2f}{\partial x\partial y}=f
\end{equation}
$$


on a Jordan domain (i.e. the interior of a simple, closed curve on the plane). The equation is hyperbolic, but we cannot formulate a Cauchy problem in the usual sense since the domain is finite, so the question is



> 
> What kind of initial value problem can we formulate on such a domain ? Which initial data along the boundary do when need to establish existence ?
> 
> 
> 



Linear hyperbolic PDE on compact two dimensional domain
Generally, you want there to be a *non-characteristic transversal*, i.e., a (let's say, smooth) curve $C$ in your domain $D$ such that each segment of each line $x=x\_0$ in $D$ is connected and meets $C$ exactly once transversely and each segment of each line $y=y\_0$ in $D$ is connected and meets $C$ exactly once transversely. Then you get existence and uniqueness for the non-characteristic initial value problem in which you specify $f$ and its normal derivative along $C$.


For example, $D$ could be a rectangle $a\le x\le b$ and $c\le y\le d$ and $C$ could be the graph of a surjective smooth map $g:[a,b]\to[c,d]$ with $g'>0$ on $[a,b]$. Solving this particular case (as Riemann did) gives the solution for any non-characteristic transversal $C$ in a domain $D$.


