# Monge Ampere equations (concavity)

The Monge-Ampere (whether real or complex, whether in a domain or on a manifold) equations usually studied are of the type $F(u, \nabla u, \mathrm{Hess}(u)) = 0$ where $F$ is a concave function of $\mathrm{Hess}(u)$. This is done for the purpose of $C^{2, \alpha}$ estimates. My question is: Has any work been done in the non-concave setting? (I know of Harvey and Lawson's work that gives us viscosity solutions, but I wish for more regular solutions) Especially with regard to the $C^{2,\alpha}$ estimate?

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Without a concavity or convexity assumption, the PDE is no longer elliptic, and regularity becomes much more difficult to prove. There is some work on this in 2-dimensions and very little in higher dimensions. –  Deane Yang Mar 23 '12 at 10:50
@Deane: the PDE could still be uniformly elliptic without being concave/convex in the Hessian of $u$. In this case you can apply the method of viscosity solutions, and if $F$ is smooth enough the best you can hope for in general is a $C^{1,\alpha}$ solution. See the book of Caffarelli-Cabre "Fully nonlinear elliptic equations" –  YangMills Mar 23 '12 at 14:51
YangMills, I stand corrected. I didn't know that. –  Deane Yang Mar 23 '12 at 15:18

Do a google search for "Fully nonlinear uniformly elliptic equations" and you will turn up a lot of information. The best estimates for viscosity solutions are $C^{1,\alpha}$, in general, where $\alpha > 0$ is very tiny. This was work in the 1980s due to Caffarelli following the important work of Krylov-Safonov. There are very recent counterexamples due to Nadirashvili and Vladut which show that this is the best you can get. The latest and greatest (arXiv: 1111.0329) has counterexamples in dimension $N \geq 5$.

If $F$ is convex or concave in the Hessian, then there are $C^{2,\alpha}$ estimates. This is the Evans-Krylov theorem. There are also estimates for certain classes of nonconvex equations with special structure (see the work of Cabre-Caffarelli).

And (ahem) if you will permit me to advertise my own work with Luis Silvestre and Charlie Smart (arxiv:1103.3677), we have a partial regularity result for a general nonlinearity, stating that the singular set is small: for some $\alpha> 0$ (very tiny) the solutions are $C^{2,1-\alpha}$ off a set of Hausdorff dimension $N-\alpha$.

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