Timeline for PDEs, boundary conditions, and unique solvability
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 27, 2010 at 23:39 | comment | added | Igor Khavkine | Umm, the broken equations should have been $\square A_i=0$, $\square A'_i=0$ and $\square u=0$. | |
| Jun 27, 2010 at 23:34 | comment | added | Igor Khavkine | Deane, I've briefly explained my motivation in the comments to Willie's first reply, though it might be impenetrable if you're not already familiar with the subject. Here's a brief example. Consider the equations $\box A_i=0$, $\box A'_i=0$, $\box u=0$, $A'_i=A_i+\partial_i u$, with both $A_i$'s and $u$ obeying sufficiently fast fall off at space-like infinity. Both conditions $\partial_z u=A'_z-A_z$ and $\nabla^2 u=\nabla\cdot(A'-A)$ are compatible with the other quations and fix $u$ uniquely for given $A$ and $A'$, but $\partial_t u=A'_t-A_t$ does not. I want alternatives to the former two. | |
| Jun 26, 2010 at 14:07 | comment | added | Deane Yang | It's probably not just the work of Ehrenpreis. For variable coefficient operators, there is the propagation of singularities (or wavefront set) of Hormander, but I'm pretty sure that there is a more refined version for constant coefficient operators. | |
| Jun 26, 2010 at 13:14 | comment | added | Willie Wong | Deane, you are probably right. I am just not familiar with his collected works (as evident from the fact that his name only brought up a single synapse firing in my brain) :p | |
| Jun 26, 2010 at 3:18 | comment | added | Deane Yang | Willie, thanks! Ultrahyperbolic is right, and so is Fritz John, not Peter Lax. I haven't looked at this since I was a graduate student many eons ago. It's the only non-elliptic non-hyperbolic non-parabolic PDE I know of whose solutions are well understood. As for the constant coefficient stuff, I don't remember that stuff either, but isn't there stuff about singularities (and maybe even the support of the Green's function) propagating only along null bicharacteristics? This might be relevant, maybe? | |
| Jun 26, 2010 at 2:28 | comment | added | Willie Wong | Ah, and thanks for jogging my memory. My only association to Ehrenpreis is that theorem where constant coefficient PDOs have Green's functions. Is there something else he did that is applicable to the problem at hand? | |
| Jun 26, 2010 at 2:22 | comment | added | Willie Wong | ... in particular it is also known as the operator from John's equation en.wikipedia.org/wiki/John's_equation . The name ultrahyperbolic, I just learned according to Wikipedia, comes from Courant. | |
| Jun 26, 2010 at 2:18 | comment | added | Willie Wong | That is the ultrahyperbolic wave operator of order (2,2). The name that I most associate with it is Fritz John. | |
| Jun 25, 2010 at 23:37 | history | answered | Deane Yang | CC BY-SA 2.5 |