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Let $f,g:\mathbb{R}^2\to\mathbb{R}$ be smooth functions and let $H_f, H_g$ be their Hessians. Is anything known about the differential equation $H_f H_g=H_g H_f$? Or, in higher dimensions, let $F=(f_1,\dots, f_n): \mathbb{R}^m\to \mathbb{R}^n$ and consider the system $H_{f_i}H_{f_j} = H_{f_j}H_{f_i}$ for all $i$ and $j$.

I've found some simple families of solutions ($f = c g$ for some constant; or $f=a(x)+b(y)$, $g = c(x) + d(y)$) but I'd like to know if there are more. I'm particularly interested in extension problems, for instance:

  • Suppose that $f$ and $g$ are defined on a line segment. When can they be extended to a solution in a neighborhood of the segment?
  • Suppose that $f$ and $g$ satisfy the equation on a neighborhood of the boundary of a circle. Can the solution be extended to the disc?

I don't know enough about PDEs to have an idea of where to start looking, so any thoughts or references would be helpful.

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    $\begingroup$ Section 6.9 of the 2nd edition of Cartan for Beginners by Ivey and Landsberg has a discussion of (and additional refernces for) the problem of commuting hessians. $\endgroup$
    – TK-421
    Commented Mar 7, 2023 at 21:59
  • $\begingroup$ Thanks! The equation seemed nice enough that I figured there must be some work on it, but my searches didn't turn up anything. I'll grab a copy of Ivey and Landsberg from the library. $\endgroup$
    – Robert Young
    Commented Mar 8, 2023 at 15:30
  • $\begingroup$ Gladly! I hope it proves helpful despite being a short section in the book. That's the only reference I know for commuting Hessians off the top of my head. It won't answer your interest in extension problems, but perhaps the additional references will lead you to relevant literature. $\endgroup$
    – TK-421
    Commented Mar 9, 2023 at 19:33

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