Timeline for Lifting a determinant map
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| S Feb 11, 2018 at 19:31 | history | suggested | Ali Taghavi |
I add two new tags because this question can be stared on an arbitrary Riemannian manifold
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| Feb 11, 2018 at 18:46 | comment | added | Will Jagy | Igor, years ago I was asking at Cody's bookstore for a periodical on soccer from England. The person helping me said "Let me find you someone wise in the ways of magazines" | |
| Feb 11, 2018 at 18:27 | review | Suggested edits | |||
| S Feb 11, 2018 at 19:31 | |||||
| Feb 11, 2018 at 18:07 | comment | added | Deane Yang | If $f$ is negative or has mixed sign, then there are results only for existence of a solution near a given point, i.e., local solutions. If $\nabla f \ne 0$, then it's known as a Tricomi-type equation, which was studied notably by Cathleen Morawetz. If $f < 0$, then you can solve for local solutions by assuming that the Hessian of $g$ has one negative eigenvalue and the rest positive and using the theory of nonlinear hyperbolic PDEs. Currently, there is absolutely no hope of finding global solutions when $f$ is not strictly positive. A radically new idea is needed. | |
| Feb 11, 2018 at 18:00 | comment | added | Deane Yang | 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. | |
| Feb 11, 2018 at 17:42 | comment | added | Igor Rivin | @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). | |
| S Feb 11, 2018 at 11:48 | history | suggested | Ali Taghavi |
I add a tag
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| Feb 11, 2018 at 11:16 | review | Suggested edits | |||
| S Feb 11, 2018 at 11:48 | |||||
| Feb 11, 2018 at 6:38 | comment | added | Deane Yang | 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. | |
| Feb 11, 2018 at 5:08 | comment | added | Igor Rivin | @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... | |
| Feb 11, 2018 at 4:06 | comment | added | Will Jagy | en.wikipedia.org/wiki/Monge%E2%80%93Amp%C3%A8re_equation | |
| Feb 11, 2018 at 3:48 | history | asked | Igor Rivin | CC BY-SA 3.0 |