Hölder estimates on solutions of non-linear elliptic PDE. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T02:09:11Z http://mathoverflow.net/feeds/question/73338 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73338/holder-estimates-on-solutions-of-non-linear-elliptic-pde Hölder estimates on solutions of non-linear elliptic PDE. semyon alesker 2011-08-21T15:58:59Z 2012-05-29T15:20:34Z <p>In his book "Some non-linear problems in Riemannian geometry" T. Aubin states the following result (Theorem 3.56):</p> <p>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, $\beta\in (0,1)$).</p> <p><strong>My question is: whether this result is true indeed, and what is the right reference?</strong></p> <p>Aubin refers to two papers by L. Nirenberg:</p> <p>(1) Comm. Pure Appl. Math.,6 (1953),103-156;</p> <p>(2) Ann. Math. Studies 33, Princeton(1954), 95-100.</p> <p>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, <strong>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?</strong></p> http://mathoverflow.net/questions/73338/holder-estimates-on-solutions-of-non-linear-elliptic-pde/73342#73342 Answer by Deane Yang for Hölder estimates on solutions of non-linear elliptic PDE. Deane Yang 2011-08-21T16:32:06Z 2011-08-21T16:32:06Z <p>I just stumbled onto this: </p> <p><a href="http://archive.numdam.org/ARCHIVE/SEDP/SEDP_1988-1989___/SEDP_1988-1989____A17_0/SEDP_1988-1989____A17_0.pdf" rel="nofollow">On fully nonlinear elliptic equations of second order by Louis Nirenberg</a></p> <p>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 <a href="http://www.math.jhu.edu/~js/msri.notes.pdf" rel="nofollow">GEOMETRIC ASPECTS OF THE THEORY OF FULLY NON LINEAR ELLIPTIC EQUATIONS by Joel Spruck</a></p> http://mathoverflow.net/questions/73338/holder-estimates-on-solutions-of-non-linear-elliptic-pde/73402#73402 Answer by Florian for Hölder estimates on solutions of non-linear elliptic PDE. Florian 2011-08-22T12:37:34Z 2011-08-22T12:37:34Z <p>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.</p>