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 ?