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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.

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    shouldn't $u_{xy} = u_{yx}$ ?2012-09-09
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    @experimentX: Well, yes, if $u$ is nice enough. I'm interested in strange $u$'s, where this does not hold.2012-09-09
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    looks like out of my scope :(2012-09-09
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    The only example I know of, where you have a function with $u_{xy}\neq u_{yx}$, the inequality is valid in a given point. I suppose it would be very difficult to build an example valid in an open set.2012-09-09
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    In the sense of distributions, $u_{xy} = u_{yx}$ always. So you won't find any interesting weak solutions.2012-09-09
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    @enzotib: Well, according to [this question](http://math.stackexchange.com/questions/104735/can-cross-partial-derivatives-exist-everywhere-but-be-equal-nowhere), it is possible to construct examples where the mixed derivatives exist everywhere and differ on a set of positive Lebesgue measure. It does indeed seem difficult to construct such things though.2012-09-09
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    I'm sure there is no general theory for this sort of thing. Linear PDE are best understood in terms of distributions (which sometimes happen to be nice functions), and as @RobertIsrael pointed out, the distributional derivatives always commute. It does make an entertaining real analysis question: if $u_{xy}+u_{yx}=0$ holds in a domain, does it follow that $u_{xy}\equiv u_{yx}\equiv 0$? I would guess that the answer is yes...2012-09-10

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