Timeline for Regularity of the solution to a differential system with variable coefficients
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| Sep 23, 2016 at 19:34 | comment | added | Deane Yang | The idea of studying a variable coefficient PDE using a nearby constant coefficient PDE is called "freezing the coefficients" and is commonly used for elliptic PDE's. It could be used for hyperbolic PDE's but it's easier just to prove the estimates directly using the variable coefficient system. There are also results known (but maybe not existence) for a PDE of real principal type. For systems, there is a paper of Nils Dencker. | |
| Sep 23, 2016 at 13:21 | comment | added | user17697 | @DeaneYang This is indeed enlightening. So let me refine a bit my question: are there some "known" classes of perturbation of a differential operator with constant coefficients $Q$ so that the perturbed operator $P$ still admits solutions? Whatever sense this question makes. Thank you | |
| Sep 23, 2016 at 13:04 | comment | added | Deane Yang | This cannot be correct. First, the solution will not necessarily be compactly supported. That's not even true when $f = 0$. Some indirect evidence is that if this heuristic were correct, then the generic system of PDE's would be solvable on a given (convex) domain. Nothing remotely close to this is known. | |
| Sep 23, 2016 at 11:38 | history | asked | user17697 | CC BY-SA 3.0 |