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Let $\Omega$ be a domain, $u$ and $f$ are real valued functions on $\Omega$, $(u_{ij})$ is the Hessian matrix of $u$. The function $f$ may change sign: that said, do there exist solutions for the following equation? $$ \det u_{ij}=f $$

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In general nothing is known. Only local solvability is known for some special cases.

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  • $\begingroup$ Thank you! Could you give me some references? $\endgroup$
    – liding
    Commented Nov 16, 2019 at 20:22
  • $\begingroup$ The first page of this paper summarizes what little is known in the 2-dimensional case. Some of this can be extended to higher dimensions. $\endgroup$
    – Deane Yang
    Commented Nov 16, 2019 at 21:55
  • $\begingroup$ Sorry, Could you give me the title of this paper? $\endgroup$
    – liding
    Commented Nov 17, 2019 at 0:39
  • $\begingroup$ Sorry about that. Here's the link: arxiv.org/abs/1003.2241. Another one is arxiv.org/abs/1204.6704 $\endgroup$
    – Deane Yang
    Commented Nov 17, 2019 at 21:22
  • $\begingroup$ Thank you! Thank you! $\endgroup$
    – liding
    Commented Nov 17, 2019 at 22:17

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