Timeline for Can the number of solutions to a system of PDEs be bounded using the characteristic variety?
Current License: CC BY-SA 4.0
11 events
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| Apr 6, 2019 at 20:04 | comment | added | Ali Taghavi | @BenMcKay Thank you very much for your comment and the references. | |
| Apr 6, 2019 at 19:29 | comment | added | Deane Yang | @AliTaghavi, please keep in mind here that both the dimension of the space of solutions and the codimension of the image are infinite dimensional. And the adjoint PDE plays no role here. You're confusing the situation here with the global theory of elliptic PDEs. | |
| Apr 6, 2019 at 18:30 | comment | added | Ben McKay | @AliTaghavi: A precise statement is difficult to make, as it requires a long discussion. There is a precise statement on p. 195 of Bryant et. al., Exterior Differential Systems, theorem 3.20, but it uses terminology that would require extensive discussion to define here. They also give the reference to Gabber's paper: O. Gabber, The Integrability of the characteristic variety, Amer. J. Math., 103, 1981, .p 445-468. | |
| Apr 6, 2019 at 18:24 | comment | added | Ali Taghavi | @BenMcKay Could you please give a reference to (and explicit formulation of ) Ofer Gabber theorem? | |
| Apr 6, 2019 at 3:20 | comment | added | Ali Taghavi | @BenMcKay however the codimension of the range is closely related to the dim of kernel by consideration of adjoint PDE. | |
| Apr 5, 2019 at 15:21 | comment | added | Ali Taghavi | @BenMcKay what about a connection between the number of points in the characteristic variety and the Codimension of the range of diff operator associated to PDE? | |
| Apr 5, 2019 at 15:08 | vote | accept | Gabe K | ||
| Apr 5, 2019 at 14:55 | comment | added | Ben McKay | The existence of the hypersurface foliation for each point in the characteristic variety does not break down for nonlinear equations. The characteristic direction in the cotangent bundle of any solution is null on the tangent vectors tangent to the leaves. But counting the number of solutions is not so clear, and depends on issues of involutivity. | |
| Apr 5, 2019 at 14:51 | history | edited | Joonas Ilmavirta | CC BY-SA 4.0 |
Fixed a typo
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| Apr 5, 2019 at 13:57 | comment | added | Gabe K | Good to know. As a follow up question, the wave equation is linear whereas the system of interest is quite non-linear (at least in the non-flat case). At the risk of asking something obvious, does this example break down if we can't take linear combinations of solutions to get a solution? | |
| Apr 5, 2019 at 13:39 | history | answered | Ben McKay | CC BY-SA 4.0 |