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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 ?

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    $\begingroup$ 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. $\endgroup$ Jul 20 at 17:21
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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 $D$.

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  • $\begingroup$ 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. $\endgroup$ Jul 22 at 7:23
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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):

Cattaneo, Alberto S.; Mnev, Pavel, Wave relations, Commun. Math. Phys. 332, No. 3, 1083-1111 (2014). arXiv:1308.5592 ZBL1300.53069.

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 $C$.

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