# Linear hyperbolic PDE on compact two dimensional domain

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 ?

• A comprehensive study of the Dirichlet problem for principal part of this operator i.e. $$\frac{\partial f}{\partial x \partial y}=0$$ is found in the references cited in this answer. Jul 20 at 17:21

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 $$D$$.
• Thank you. Is it possible to extend Riemann's method to more general domains ? For instance, on a concave domain ? In that case some segments of $x=x_0$ or $y=y_0$ will not be connected. Jul 22 at 7:23
Section 4 of the following paper considers in some detail the 2D wave equation ($$\partial_x\partial_y f = 0$$ in your coordinates; not exactly the same but closely related) on compact domains with smooth boundary (with unpleasant cases where the boundary is too closely tangent to null/characteristic directions is also excluded):
A quick summary that doesn't do justice to all the details: The wave equation has a variational formulation, which indirectly means that the boundary data (value and normal derivative of $$f$$ on the boundary) gets an induced symplectic structure. The authors show that the restriction of solutions $$f$$ to the boundary gives a Lagragian subspace $$L$$ of this space of boundary data. Thinking abstractly, any set of boundary conditions identifies another (affine) subspace $$C$$ of the boundary data. The boundary conditions specify a unique solution when the two subspaces are complementary, the intersection $$L\cap C$$ is zero dimensional (just one point, the unique solution). The authors then consider some examples of such subspaces $$C$$. Heuristically (ignoring the infinite dimensional context), a generic second Lagrangian subspace would give a good $$C$$.