Timeline for Reference request: Schauder estimate in the space variable for parabolic equations
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 13, 2019 at 14:55 | vote | accept | Yuxiao Xie | ||
| Dec 12, 2019 at 5:23 | comment | added | Yuxiao Xie | I tried and I think it works: Take a normal coordinate neighborhood $U$ and a cutoff function $\zeta$ supported in $U$. We only need to estimate the $C^\alpha$ norm of $(\Delta-\partial_t)(\zeta u)$ in the space variable, where $\Delta$ is the standard Laplacian on $\mathbb{R}^n$. Then the proof is routine, using some basic interpolation inequalities for Hölder spaces. This is pretty much the same proof as the standard Schauder estimate, obtained by freezing the coefficients. However I'm not an expert at PDE so I'm not sure whether I've made any serious mistakes here. | |
| Dec 11, 2019 at 19:31 | comment | added | Andrew | I think that probably yes, but don't know how exactly. May be it would bу easier to obtain estimates for volume potential. Since the issue of smoothness is local it boils down to estimates of the volume potential in $\mathbb R^n$ for a uniformly parabolic operator with smooth coefficients. In $\S11$ of the same chapter such estimates for Holder $f$ are obtained. One has to check that no smoothness wrt $t$ is used when the required estimates $x$ are derived. | |
| Dec 11, 2019 at 16:26 | comment | added | Yuxiao Xie | Thanks! If this is true, could it be possible that the manifold case can be obtained by "freezing the coefficients"? | |
| Dec 11, 2019 at 14:41 | history | answered | Andrew | CC BY-SA 4.0 |