Timeline for Solving PDE with Cauchy - Kowalewski Theorem
Current License: CC BY-SA 3.0
9 events
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| Oct 3, 2012 at 15:48 | comment | added | Robert Bryant | @Willie: Yes, but that is the problem; the expanded expressions will involve $\beta$, and these 'integrability conditions' will define a locus in $(x,y,\beta)$-space that contains the only possible graphs of solutions. One still has to examine that zero locus to see whether or not it contains any graphs of solutions, and it may or it may not. One must also check the other integrability conditions, which are $\partial_{x^j}F_i = \partial_{x^i}F_j$ and $\partial_{y^j}G_i = \partial_{y^i}G_j$ and which must also be expanded out to get expressions to constrain the locus in $(x,y,\beta)$-space. | |
| Oct 3, 2012 at 15:12 | comment | added | Willie Wong | @Robert: can one not plug in the equation there? $$\partial_{y^k}(F_j(x,y,\beta)) = (\partial_{y^k}F_j)(x,y,\beta) + (\partial_\beta F_j)(x,y,\beta)G_j(x,y,\beta)$$ Of course this still depends on the unknown $\beta$, but if the integrability condition is only satisfied for isolated values of $\beta$ one may expect trouble. | |
| Oct 3, 2012 at 15:06 | vote | accept | Andrei | ||
| Oct 3, 2012 at 14:22 | comment | added | Robert Bryant | @Willie: You have to be a bit careful with the integrability conditions. Since $F$ and $G$ involve the unknown $\beta$, you can't just check whether $\partial_{y^k}F_j=\partial_{x^j}G_k$, since expanding these out will involve the partials of the unknown $\beta$, and you won't know whether you have equality until you know the solution $\beta$ that you are hoping to find, since these equations only have to hold for the solution $\beta$ that also satisfies the given initial conditions (assuming that it actually exists). | |
| Oct 3, 2012 at 14:15 | answer | added | Robert Bryant | timeline score: 10 | |
| Oct 3, 2012 at 7:19 | comment | added | Willie Wong | Deane Yang's paper "Local Solvability of Overdetermined Systems Defined by Commuting First-Order Differential Operators" may help (1986, CPAM), I'll see if I can get him to say a few words here. | |
| Oct 3, 2012 at 7:18 | comment | added | Willie Wong | If you are specifying the partial derivatives, you need necessarily compatibility conditions. As stated the PDE can be overdetermined. Do you know that $\partial_{y_k} F_j = \partial_{x_j} G_k$ for example (they need to be since partial derivatives commute on $\beta$)? Also, typically Cauchy-Kowelewski is used to solve a "initial value problem" where the data is prescribed on a co-dimension 1 set. You need something closer to Cartan-Kahler. | |
| Oct 3, 2012 at 7:03 | history | edited | Willie Wong |
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| Oct 3, 2012 at 6:22 | history | asked | Andrei | CC BY-SA 3.0 |