Hölder estimates on solutions of non-linear elliptic PDE. - MathOverflow

Hölder estimates on solutions of non-linear elliptic PDE.

2011-08-21T15:58:59Z

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?

Answer by Deane Yang for Hölder estimates on solutions of non-linear elliptic PDE.

2011-08-21T16:32:06Z

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

Answer by Florian for Hölder estimates on solutions of non-linear elliptic PDE.

2011-08-22T12:37:34Z

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.