Hölder estimates for the Complex Monge-Ampere equation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T08:45:36Z http://mathoverflow.net/feeds/question/92319 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92319/holder-estimates-for-the-complex-monge-ampere-equation Hölder estimates for the Complex Monge-Ampere equation Vamsi 2012-03-26T22:58:00Z 2012-06-26T13:46:56Z <p>If on a bounded smooth, pseudoconvex domain in $\mathbb{C}^n$, $\mathrm{det} ( \mathrm{Hess}(u)) = f$ ($f>0$, $\mathrm{Hess}(u)>0$, $u=0$ on the boundary), if $f \in C^{k, \alpha}$, is $u \in C^{k+2, \alpha}$ ? (I mean, is there an apriori estimate on $u$ with the Hölder exponents of $f$ and $u$ being the same (equal to $\alpha$?)</p> http://mathoverflow.net/questions/92319/holder-estimates-for-the-complex-monge-ampere-equation/93171#93171 Answer by YangMills for Hölder estimates for the Complex Monge-Ampere equation YangMills 2012-04-04T22:07:15Z 2012-06-26T13:46:56Z <p>I assume that by $\text{Hess}(u)$ you mean the complex Hessian of $u$, so $u$ is plurisubharmonic. If $k\geq 1$ then you can differentiate the equation and apply Evans-Krylov to get uniform interior $C^{k+2,\alpha}$ estimates for $u$.</p> <p>However if $k=0$ the situation is much more delicate. Assume $f$ is $C^\alpha$ and positive and $u$ is $C^2$, say, and suppose you want to prove uniform interior $C^{2,\beta}$ bounds for $u$ (here $\beta&lt;\alpha$). Then the best result to date, due to <a href="http://arxiv.org/pdf/1111.0902.pdf" rel="nofollow">Y. Wang</a>, says that you have such estimates but depending also on a pointwise upper bound for $\Delta u$. There are earlier results of <a href="http://arxiv.org/pdf/1004.0543.pdf" rel="nofollow">Chen-He,</a> <a href="http://arxiv.org/pdf/1005.0939.pdf" rel="nofollow">Blocki-Dinew</a> and <a href="http://arxiv.org/pdf/1006.4261.pdf" rel="nofollow">Dinew-Zhang-Zhang.</a> Some of these results get the same exponent $\alpha$ instead of $\beta&lt;\alpha$, under stronger assumptions.</p> <p>It is an open problem to show that if $u$ is just $L^\infty$ and plurisubharmonic, and $u$ solves the Dirichlet problem for the complex Monge-Amp&egrave;re equation in the sense of Bedford-Taylor, with RHS $f$ positive and $C^\alpha$, then $u$ is indeed $C^{2,\alpha}$. This would be the direct analogue of a well-known result of Caffarelli for the real Monge-Amp&egrave;re equation, but it is still open. An old paper of Schulz claims this result, but it contains a mistake.</p>