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I'm researching on discrete/semi-discrete/smooth differential geometry. Recently, I transformed one of my surface theory problems (in the smooth scenario) into the following Moutard PDE

$$h_{uv} = q\,h$$

where $h(u,v)$ and $q(u,v): [0,a]\times [0,b] \subset \mathbb{R}^2 \rightarrow \mathbb{R}$ are smooth functions on $(0,a)\times (0,b)$. I have the functions $h(u,0)$ and $h(0,v)$ and $q(u,v)$ as well. By observing its discrete/semi-discrete counterparts I believe that it should have a unique solution by the boundaries that I provided. However, since my knowledge of PDE is very limited I don't know how to solve it in the smooth case.

I would very much appreciate if you can direct me on how to solve it. I would specially like to know if $h(u,0)$ and $h(0,v)$ being smooth is enough for getting the unique solution. Or should they be specifically analytic?

Extra Info: In the special case that I am considering $q(u,v) = \cos(\omega(u,v))$ where $\omega(u,v)$ is a solution of the famous sine-Gordon equation:

$$\omega_{uv} = \sin{\omega}$$

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  • $\begingroup$ Is that really all you know about $q$? If you have more information about $q$ it might make it easier. $\endgroup$
    – ResearchMath
    Commented May 22 at 15:37
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    $\begingroup$ Classically this type of boundary value problem is called a "Goursat problem" (the boundaries where the data is provided are so-called characteristic curves (codim-1 surfaces really), as pointed out by Deane). Some quick searching for "2d Goursat problem" turns up this chapter from a book by Rubinstein & Rubinstein (1994). $\endgroup$
    – Igor Khavkine
    Commented May 23 at 10:29
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    $\begingroup$ @IgorKhavkine I think you pointed me to a very helpful reference. Thanks for the 2nd time helping me with my PDEs. $\endgroup$
    – RWien
    Commented May 23 at 13:15
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    $\begingroup$ By $u=c$, I mean any line parallel to $u=0$. $c$ is just an arbitrary constant. As @IgorKhavkine points out these lines are characteristic "surfaces", so standard initial value problems do not apply. $\endgroup$
    – Deane Yang
    Commented May 23 at 18:58
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    $\begingroup$ A discussion of how to solve using inverse scattering techniques the Goursat problem for the sine-Gordon equation can be found here: epubs-siam-org.proxy.library.nyu.edu/doi/abs/10.1137/0134004 $\endgroup$
    – Deane Yang
    Commented May 23 at 18:59

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