Timeline for Smooth Cauchy problem on a cylindrical manifold (or how to define the exponential of a differential operator)
Current License: CC BY-SA 4.0
7 events
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| Oct 9, 2018 at 19:25 | comment | added | user209029 | @IgorKhavkine , thank you!! The Chernoff paper is exactly what I need for what I'm working on! But I think I'm going to leave the question open, just in case somebody wants to contribute with a more general theorem (although I'm starting to doubt it's even possible). | |
| Oct 8, 2018 at 20:15 | comment | added | Deane Yang | I believe that such a problem is well posed only if $\partial_t - D$ is hyperbolic. In other words, if $a^i\partial_i$ is the first order term for $D$, then for each $x \in M$, the roots $\xi$ of the polynomial $p(x, \xi) = \det a^i(x)\xi_i$ are all real. However, this condition is in general not sufficient. It is known to be sufficient under certain conditions. The best known are symmetric hyperbolic, where $a^i$ are symmetric matrices, and strictly hyperbolic, where the roots of $p$ are real and distinct. | |
| Oct 8, 2018 at 19:16 | comment | added | Igor Khavkine | 3) Though sometimes (Chernoff 1973). | |
| Oct 8, 2018 at 19:09 | comment | added | Igor Khavkine | 3) Often not (en.wikipedia.org/wiki/Heat_kernel). To get an idea of what's involved in constructing $e^{tD}$ using the spectral theory of $D$ (at least in the case when it has self-adjoint closed extensions), have a look at this old blog post by Terry Tao: terrytao.wordpress.com/2011/12/20/… | |
| Oct 8, 2018 at 16:03 | comment | added | Ben McKay | If you allow 2nd order $D$, you can get the Schrodinger equation, and then there are many books written about the problem of scattering theory for those, i.e. your problem is too hard to have a general solution. | |
| Oct 8, 2018 at 15:25 | review | First posts | |||
| Oct 8, 2018 at 15:59 | |||||
| Oct 8, 2018 at 15:23 | history | asked | user209029 | CC BY-SA 4.0 |