This is a kind of a follow-up to Question on Hessian of a function (probability question). Suppose I give you a continuous function $f:\mathbb{R}^n \to \mathbb{R}.$ Is it true that there exists a ($C^2$) function $g:\mathbb{R}^n \to \mathbb{R},$ such that $\det(\mbox{Hess}(g)) = f?$ (where $\mbox{Hess}$ is the Hessian, of course)?
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asked Feb 11, 2018 at 3:48
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3$\begingroup$ en.wikipedia.org/wiki/Monge%E2%80%93Amp%C3%A8re_equation $\endgroup$– Will JagyCommented Feb 11, 2018 at 4:06
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2$\begingroup$ @WillJagy Truly you are wise in the ways of science! However, the question is somewhat less general than the general Monge-Ampere equation, in that $f$ only depends on $x$ in my question, and not on $u, u^\prime,$ so that should be easier, I should think... $\endgroup$– Igor RivinCommented Feb 11, 2018 at 5:08
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3$\begingroup$ If you don’t restrict $f$ to be a positive function, then very little is known. If $f > 0$, you can try to solve for a convex solution Then the PDE is elliptic, so there is hope. I’m not sure what is known for a Monge-Ampère equation on all of $\mathbb{R}^n$. But your colleague Cristian Gutierrez is an expert on this. $\endgroup$– Deane YangCommented Feb 11, 2018 at 6:38
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1$\begingroup$ @DeaneYang I read Cristian's book (ok, skimmed) yesterday, and, as you say, it's all about positive $f,$ and even then I cannot figure out if you can ever get $C^2$ regularity. The case of interest for the referenced question is the sign-variable question, about which nothing is known, and I can't even find too much on the hyperbolic Monge-Ampere (and Cristian does not mention it). $\endgroup$– Igor RivinCommented Feb 11, 2018 at 17:42
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1$\begingroup$ For the positive $f$ case, elliptic theory gives only Holder estimates and never exactly $C^2$. So the best you can hope for is that if $k \ge 0$, $0 < \alpha < 1$, and $f$ is $C^{k,\alpha}$, then $g$ is $C^{k+2,\alpha}$. I say "hope for", because I don't know what exactly is known. $\endgroup$– Deane YangCommented Feb 11, 2018 at 18:00
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