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In his book "Some non-linear problems in Riemannian geometry" T. Aubin states the following result (Theorem 3.56):

Let $A(u)=F(x,u,\nabla u,\nabla^2u)$ be a non-linear second order differential operator defined on an open subset $\Omega$ of $\mathbb{R}^n$; $F$ is an infinitely smooth function. Let $\Theta$ be a bounded subset of $C^2(\Omega)$. Suppose $A$ is uniformly elliptic on $\Omega$ uniformly in $u\in \Theta$. Then if $A(\Theta)$ is bounded in $C^{r,\beta}(\Omega)$, then $\Theta$ is bounded in $C^{r+2,\beta}(K)$ for any compact subset $K\subset \Omega$ (here $r\geq 1$ is an integer, $\beta\in (0,1)$).

My question is: whether this result is true indeed, and what is the right reference?

Aubin refers to two papers by L. Nirenberg:

(1) Comm. Pure Appl. Math.,6 (1953),103-156;

(2) Ann. Math. Studies 33, Princeton(1954), 95-100.

Also Aubin refers to previous results by other people containing some weaker statements. I do not have the 1954 paper (and it does not contain the detailed proof in fact), but the 1953 paper deals only with the case of plane $n=2$. I need the case $n>2$. In the 1953 paper Nirenberg mentions that he has generalized his result from $n=2$ to higher dimensions, but in a somewhat weaker form. If I understand correctly, in addition he needs a bound on the modulus of continuity of second derivatives of functions from the set $\Theta$. Was this assumption removed since than?

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2 Answers

I just stumbled onto this:

On fully nonlinear elliptic equations of second order by Louis Nirenberg

which seems to answer your question in the affirmative. Also, are you really studying a general fully nonlinear PDE or are you studying a particular one such as Monge-Ampere? There is a lot more known about the latter. Another set of notes that I have not read but look interesting are GEOMETRIC ASPECTS OF THE THEORY OF FULLY NON LINEAR ELLIPTIC EQUATIONS by Joel Spruck

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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. –  semyon alesker Aug 22 '11 at 7:32
    
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. –  semyon alesker Aug 22 '11 at 7:33
    
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. –  Deane Yang Aug 22 '11 at 13:42
    
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. –  Deane Yang Aug 22 '11 at 13:43
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@ 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. –  S.A.A Nov 10 '12 at 4:00
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Gilbarg and Trudinger, "Elliptic partial differential equations of second order" (1998 or 2001), Lemma 17.16 seems to be the result you are looking for. It does not contain a statement about the uniformity of the estimates, but this should follow from the proof.

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@Florian: I did not check the details of the proof of Lemma 17.16. But I am afraid that they may use somewhere modulus of continuity of second derivatives of a solution (for a single $C^2$ function it always exists). –  semyon alesker Aug 22 '11 at 12:50
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