I was thinking a bit about PDEs and realized that I haven't seen any PDEs whose solutions possess non-equal mixed partial derivatives or where this possibility is at least taken seriously. So, I was wondering:
What is known about such equations? Is there a general theory of them?
A reference would be very welcome. (If such equations make sense, of course.)
In particular, I was wondering what can be said about the equation
$$u_{xy}+u_{yx}=0$$
Are there any interesting solutions to this equation, in any sense?
I am mostly interested in functions $u:\Omega\to\Bbb R$, where $\Omega\subseteq\Bbb R^2$ is a domain and preferably $u_{xy}\neq 0$, but weak solutions of any kind are also welcome.
Thanks in advance.