Timeline for Does Hölder continuity imply smoothness for the CMC equation: $u:D^2\rightarrow\mathbb{R}^n$, $\Delta u = 2H\partial_xu\times\partial_yu$, $H$ constant?
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7 events
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| Feb 10, 2011 at 22:04 | vote | accept | Glen Wheeler | ||
| Feb 10, 2011 at 22:04 | comment | added | Glen Wheeler | You're right. Thanks for the answer :). | |
| Feb 10, 2011 at 19:10 | comment | added | Willie Wong | (In fact that paper also contains a proof of Holder regularity from just $W^{1,2}$...) | |
| Feb 10, 2011 at 19:08 | comment | added | Willie Wong | The reference I gave is to obtain $C^{1,\alpha}$ from just Holder. I made that comment since in your question you ask for smooth, while the section title for section 3 of Chang, Wang, and Yang's paper is "$C^{1,\gamma}$ regularity", so I thought I'd remark that that is enough. | |
| Feb 10, 2011 at 18:48 | comment | added | Glen Wheeler | Thanks Willie. I will check out the reference. But in this question I only have Hoelder continuity of the solution, not $C^{1,\alpha}$. I am aware that once the RHS is $C^\alpha$ then the standard regularity argument works fine. Thanks for the reference though, if one needs to work harder and obtain $C^{1,\alpha}$ then that's still useful. On Helein: I have been through Helein's book. He obtains the Hoelder continuity of the solution and then refers to the elliptic book by Ladyzhenskaya and Uraltseva. I couldn't find the argument with the correct growth on the RHS in that book however. | |
| Feb 10, 2011 at 17:29 | history | edited | Willie Wong | CC BY-SA 2.5 |
added 319 characters in body
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| Feb 10, 2011 at 17:23 | history | answered | Willie Wong | CC BY-SA 2.5 |