Timeline for Hölder estimates on solutions of non-linear elliptic PDE.
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Nov 10, 2012 at 4:08 | comment | added | Deane Yang | S.A.A., thanks for correcting my remarks! | |
| Nov 10, 2012 at 4:00 | comment | added | S.A.A | @ Semyon: Yes, before Evans-Krylov theory (like the case of Aubin and Yau's original proof) they needed $C^3$. Having $C^{2,\alpha}$ you can differentiate the equation and prove higher order estimates. | |
| Aug 23, 2011 at 7:59 | comment | added | asv | Deane, as far as I understand $C^{2,\alpha}$ does imply $C^{k,\alpha}$ under rather general assumptions. This is exactly what I am trying to understand now in some detail. However the people in Monge-Ampere equations often work hard to get $C^3$ estimate because it may be not simpler than to get $C^{2,\alpha}$. Correct me if I am wrong, but apparently the method of Evans-Krylov did change the situation: it allows to get $C^{2,\alpha}$ from $C^2$ in some cases. | |
| Aug 22, 2011 at 13:43 | comment | added | Deane Yang | Also, I don't believe that $C^{2,\alpha}$ implies $C^{k,\alpha}$, because people devote a lot of effort to prove a $C^3$ estimate for the real and complex Monge-Ampere equation. | |
| Aug 22, 2011 at 13:42 | comment | added | Deane Yang | Semyon, I forgot that you were working on this. I suspect that you need to take advantage of the special form of the equation and need more than just the general theory of fully nonlinear elliptic PDE's. You probably need to adapt the ideas and techniques developed for the complex Monge-Ampere equation. | |
| Aug 22, 2011 at 7:33 | comment | added | asv | Deane, I am interested in Monge-Ampere equations, but in quaternionic ones. They are not discussed much in literature. They can be written on a general hypercomplex manifold (whatever that means...). If the hypercomplex structure is locally flat then the above condition of convexity of $F$ is satisfied, I think. I have not checked the general case. | |
| Aug 22, 2011 at 7:32 | comment | added | asv | Thanks, Deane! Indeed this partly answers my question. In this paper by Nirenberg you mentioned $C^2$ estimate implies $C^{2,\alpha}$ estimate under an extra assumption of some convexity of the function $F$ . This assumption was not mentioned in the Aubin's book. It is satisfied in my situation, at least in some cases (I have not checked the general case- see my next comment). However I still do not see under what conditions $C^{2,\alpha}$ estimate implies higher $C^{k,\alpha}$ . Probably it is even more classical, so Nirenberg does not discuss this at all. | |
| Aug 21, 2011 at 16:32 | history | answered | Deane Yang | CC BY-SA 3.0 |