Timeline for Upper bound Hölder norm of the solution to the linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (x)|^2 u(t, x) \}$
Current License: CC BY-SA 4.0
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| Feb 21 at 21:19 | comment | added | Connor Mooney | The function $v := \sigma^2u$ solves $\partial_t v = \sigma^2\Delta v$. Parabolic Schauder estimates (e.g. from the reference above or Lieberman's book) give $C^{3,\,\alpha}$ regularity of $v$, hence $C^{1,\alpha}$ regularity of $u$. | |
| Feb 15 at 12:37 | comment | converted from answer | Robert Wegner | I think the following document might be interesting for you: people.math.harvard.edu/~spicard/notes-parabolicpde.pdf It contains many estimates for a class of parabolic equations which I think includes yours. They use a kind of space-time Hölder space but it should be possible to extract some results from those which are similar to what you are looking for. | |
| Feb 15 at 6:18 | history | asked | Akira | CC BY-SA 4.0 |