Timeline for Inverse of partial differential operator as a smooth tame map
Current License: CC BY-SA 3.0
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| when toggle format | what | by | license | comment | |
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| Jun 21, 2016 at 18:30 | comment | added | Pedro Lauridsen Ribeiro | I did that, he deals with the homogeneous Cauchy problem only (i.e. without sources). The point is that the loss of derivatives is not that severe for the fixed-point iteration scheme aiming at solving the homogeneous Cauchy problem to work, that's why choosing appropriate norms and interpolating with Gagliardo-Nirenberg saves the day. | |
| Jun 21, 2016 at 10:24 | comment | added | Deane Yang | Pedro, I suggest consulting the papers of Klainerman written after the one I cited. The point is that you can use the energy estimates, the Gagliardo-Nirenberg inequalities, and the "right" functional norms (which distinguish the time coordinate from the space coordinates), to get the estimates needed for the Banach space implicit function theorem. | |
| Jun 20, 2016 at 13:29 | comment | added | Pedro Lauridsen Ribeiro | Well Deane, if a (say) quasilinear hyperbolic PDE has a source term on the right hand side $$P(\phi)=g(\phi,\partial\phi)\partial^2\phi+H(\phi,\partial\phi)=f\ ,$$ I don't see any obvious way of avoiding using Nash-Moser and solving it with a contraction mapping argument, and that is where you really have to deal with local invertibility - typically, $f$ has the form $f=P(\phi_0)+\tilde{f}$ with $\tilde{f}$ "small". You had to face such a problem in that DMJ paper of yours with Robert Bryant and Phillip Griffiths on low-dimensional isometric embeddings. | |
| Oct 12, 2014 at 17:33 | vote | accept | Tobias Diez | ||
| Sep 17, 2014 at 18:51 | history | answered | Deane Yang | CC BY-SA 3.0 |