TITLE: General solution to an ultrahyperbolic PDE
QUESTION [5 upvotes]: $\DeclareMathOperator\SO{SO}$The following PDE defined on $\mathbb{R}^2$ $$\frac{\partial}{\partial x}\frac{\partial}{\partial y}f(x,y) = 0,$$ has solution $$f(x,y) = g(x) + h(y),$$ where $g,h : \mathbb{R} \to \mathbb{R}$ are arbitrary (nice enough) functions. I believe this to be a general solution to the equation, for any choice of boundary conditions. I don't have a proof of this and I'm happy to be shown otherwise if that's the case.
I want to solve a similar looking PDE for $(x,y) \in \mathbb{R}^{n}\times\mathbb{R}^{n},$ which is given by $$\frac{\partial}{\partial x}\cdot\frac{\partial}{\partial y}f(x,y) = 0,$$ where the dot product is with the Euclidean metric on $\mathbb{R}^{n}$. There is an $\SO(n)$ symmetry under $x \mapsto M x, y \mapsto M^T y$. I can see that there are still solutions of the form $f_1(x,y) = g(x) + h(y)$ where $g,h : \mathbb{R}^n \to \mathbb{R}$ , but that these are no longer the only solutions.
I've come up with something quite general that solves the equation;$$f_2(x,y) = \int d^na\,\left( g^x(x \wedge a) \,h^y(y\cdot a) + g^y(y \wedge a) \,h^x(x\cdot a)\right) + f_1(x,y),$$ where   $g^x,g^y:\mathbb{R}^n\times\mathbb{R}^n\to \mathbb{R}$ and $h^x,h^y:\mathbb{R} \to \mathbb{R}$ are arbitrary (sufficiently nice) functions, which can depend on $a$. I added on $f_1(x,y)$ explicility, although I think it may be possible to fiddle the form of the integral and the choice of $g^x,g^y,h^x,h^y$ to include solutions of the form $f_1(x,y)$ without adding them on separately. One feature that I like about this solution is that it respects the $\SO(n)$ symmetry of the original PDE. The integral over the vector $a$ is also natural, in the context of the physical problem where the PDE comes from.
Is anyone able to find a general solution to this equation, in a form which respects the $\SO(n)$ symmetry, possibly with some (hopefully quite general) restriction on the boundary conditions?
Is my solution general for some choice of boundary conditions?

REPLY [2 votes]: This is not an answer to the main question as the one offered by Robert Bryant, but it is a comment to the statement pertaining the two-variable boundary value problem for the ultrahyperbolic equation:

I believe this to be a general solution to the equation, for any choice of boundary conditions. I don't have a proof of this and I'm happy to be shown otherwise if that's the case.

The Dirichlet problem
$$
\begin{cases}
\dfrac{\partial}{\partial x}\dfrac{\partial}{\partial y} f(x,y)=f_{xy}(x,y)=0 & x,y\in[0,1]\\
f(x,\alpha(x))=\varphi_1(x) & x\in [0,1]\\
f(x,\beta(x))=\varphi_2(x) & x\in [0,1]
\end{cases}\label{1}\tag{1}
$$
where $\alpha, \beta:[0,1] \to [0,1]$ are appropriate strictly monotonic functions, is overdetermined (this, according to [1] pp. 6, was proved by Mauro Picone in [3]). Problem \eqref{1} was thoroughly studied by Gaetano Fichera: in [2] he found necessary and sufficient conditions (compatibility conditions) on the Dirichlet datum $(\varphi_1,\varphi_2)$ for it to be solvable with $f\in C^1([0,1]^2)$. In [1] he drops these compatibility conditions in order to accurately describe the singularities developed by the general solution
$$
f(x,y) =g(x)+ h(y),\label{2}\tag{2}
$$
by describing the kind of singularities developed by $g(x)$ and $h(y)$ as $x,y\to 0^+$. Therefore, it is probably right that \eqref{2} is the general solution to \eqref{1}, even when the boundary data are such that the solution behaves singularly at some point of the domain where the problem is posed.
References
[1] Gaetano Fichera, "Studio delle singolarità della soluzione di un problema di Dirichlet per l'equazione $u_{xy} = 0$" (Italian), Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Serie 8 50 (1971), fasc. n.1, pp. 6-17, MR0310434, Zbl 0231.35046.
[2] Gaetano Fichera, "Su un problema di Dirichlet per l'equazione $u_{xy}=0$". (Italian) Atti dell'Accademia delle Scienze di Torino, Classe di Scienze Fisiche Matematiche e Naturali 105 (1971), pp. 355–366, MR298228.
[3] Mauro Picone, "Sulle equazioni alle derivate parziali del second’ordine del tipo iperbolico in due variabili indipendenti" (Italian), Rendiconti del Circolo Matematico di Palermo 30, pp. 349-376 (1910), DOI: 10.1007/BF03014882, JFM 41.0424.04.