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We recently realized that a geometric questions of interest to us is strongly related to the regularity of solutions of the following simple equation on the unit disk in $R^2$: $$ \det(Hess(w))=1~, $$ where $w$ is a complex-valued function.

Clearly this could be considered as a Monge-Ampère equation, but it's not what people usually call a complex Monge-Ampère equation. It can be written in terms of the real and imaginary parts of $w$ but it then doesn't look as nice, and the imaginary part of the equation mixes the real and imaginary parts of $w$ in a fairly nasty way.

We would be grateful for any pointer to studies of related PDEs.

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    $\begingroup$ If you don't put additional constraints on $w$, regularity could potentially be bad. You can check that $w = i xy$ is a solution to your equation, and if you linearise the equation around it, the equation looks like $xy w_{xy} = 0$ which is, where it is not degenerate, hyperbolic. So I wouldn't hold up too much hope for a general regularity theory. $\endgroup$
    – Willie Wong
    Nov 22, 2013 at 13:53
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    $\begingroup$ Come to think of it, isn't your equation solved by any function $w = u + i xy$ where $u = u(y)$? This already shows that the regularity can be as bad as your notion of solution (classical/weak/etc.) allows. $\endgroup$
    – Willie Wong
    Nov 22, 2013 at 13:57
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    $\begingroup$ If the real part is strictly convex, then the imaginary part solves a linear elliptic equation with variable coefficients. If we consider the imaginary part fixed, the real part in fact solves a Monge Ampere equation. So playing a bit with the standard machinery would probably give you already some reasonable regularity results. $\endgroup$
    – Willie Wong
    Nov 22, 2013 at 16:12
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    $\begingroup$ Could you say more about what you're looking for? As Willie points out, if $u$ and $v$ are assumed to have positive definite Hessians, this is a second order nonlinear elliptic system. This implies local existence of solutions. And there are minimal regularity conditions on $u$ and $v$ that imply $u$ and $v$ are smooth. $\endgroup$
    – Deane Yang
    Nov 26, 2013 at 13:31
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    $\begingroup$ Moreover, if you start with a sufficiently smooth solution, say, $$ u = \frac{1}{2}(x^2 + y^2) \text{ and }v = 0, $$ on a the unit disk, then you can use the implicit function theorem and the solution to the linear Dirichlet problem to show the existence of nearby solutions with perturbed boundary data. This would also show uniqueness of the Dirichlet problem for boundary data near the initial boundary data. $\endgroup$
    – Deane Yang
    Nov 26, 2013 at 13:31

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