Timeline for Boundary behaviour of a second order pde with characteristics
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 10, 2015 at 9:06 | vote | accept | guido giuliani | ||
| Dec 10, 2015 at 9:06 | comment | added | guido giuliani | @DeaneYang Nope... they do not play any particular role. | |
| Dec 9, 2015 at 22:33 | comment | added | Deane Yang | Do $Y$ and $Z$ play any role here? | |
| Dec 9, 2015 at 21:53 | answer | added | Bazin | timeline score: 2 | |
| Dec 9, 2015 at 14:12 | comment | added | Robert Bryant | In the case of this specific vector field, the differentiable solutions of your equation are of the form $g(x,y,z) = p_0(y,2z{+}xy)+xp_1(y,2z{+}xy)$ where $p_0(u,v)$ and $p_1(u,v)$ are differentiable functions of their arguments, and if $\pi_0$ and $\pi_1$ are disjoint surfaces transverse to $X$ such that each of them meets each integral curve of $X$ exactly once, then knowing $g$ on each of $\pi_0$ and $\pi_1$ will uniquely determine $p_0$ and $p_1$. I don't have a reference in mind for the counterexample in the general case, just a counterexample. | |
| Dec 9, 2015 at 10:29 | comment | added | guido giuliani | @RobertBryant I've specified which vector field $X$ I am concerned with. However your answer already contains an interesting point about existence of solution. May I ask for any particular reference that you have in mind? Best regards. | |
| Dec 9, 2015 at 10:27 | history | edited | guido giuliani | CC BY-SA 3.0 |
added 260 characters in body
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| Dec 9, 2015 at 10:06 | comment | added | Robert Bryant | You need to specify what you mean by 'play the same roles'. Just requiring $X$ to be transverse to both $\pi_0$ and $\pi_1$ is not sufficient in general. You need to know more about the dynamics of the vector field $X$ before you can assert that there exists a solution with specified values on $\pi_0$ and $\pi_1$. | |
| Dec 9, 2015 at 9:52 | history | asked | guido giuliani | CC BY-SA 3.0 |