Timeline for Sobolev spaces and geometry
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 26, 2023 at 2:04 | comment | added | Willie Wong | @PieroD'Ancona I don't think I fully agree with your last comment, especially in view of the answer you gave. On $\mathbb{R}^n$ we can define $W^{k,p}$ by (a) looking at $L^p$ functions that are $k$-times weak differentiable, with weak derivatives in $L^p$; (b) Look at $C^\infty \cap L^p$ and complete it using the $W^{k,p}$ norm; (c) Look at $C^\infty_c$ and complete it using the $W^{k,p}$ norm. They give you the same thing on $\mathbb{R}^n$. On manifolds, without bounds on geometry at infinity, these three definitions can give different spaces. | |
| Mar 5, 2014 at 23:03 | comment | added | Piero D'Ancona | Sobolev norms measure regularity and size of functions. Regularity is a local thing, so it does not really have much to do with geometry. Size has in part to do with the behaviour of the geometry at infinity, but again, I think the contact with geometry is not really essential. | |
| Mar 5, 2014 at 21:36 | comment | added | Juan OS | Thanks for your answer! So I guess you might say that Sobolev spaces might "measure" how complicated the energy functional is in some manifold? or is that not at all what you mean? | |
| Mar 5, 2014 at 21:31 | comment | added | Piero D'Ancona | It's the farthest from PDEs I managed to get :) | |
| Mar 5, 2014 at 21:02 | comment | added | Tim Seguine | That connection with the Dirichlet functional brings one naturally to the calculus of variations, which sort of brings one full circle back to the study of PDEs, or am I babbling? | |
| Mar 5, 2014 at 18:12 | history | answered | Piero D'Ancona | CC BY-SA 3.0 |