The MongeAmpere (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 nonconcave 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?

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 KrylovSafonov. 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 EvansKrylov theorem. There are also estimates for certain classes of nonconvex equations with special structure (see the work of CabreCaffarelli). 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 $N\alpha$. 


The classical Minkowski problem is that of existence, uniqueness and regularity of closed convex hypersurfaces of the Euclidean linear space R^(n+1) whose Gaussian curvature (in the sense of Gauss’ definition) is prescribed as a function of the outer normal vector. For C2+ hypersurfaces (C2hypersurfaces with positive Gaussian curvature), this Minkowski problem is equivalent to the question of solutions of certain Monge–Ampère equations of elliptic type on the unit sphere S^n of R^(n+1). This classical Minkowski problem has a natural extension to hedgehogs, that is to Minkowski differences H = K − L of closed convex hypersurfaces K, L in R^(n+1), at least if we restrict ourselves to hypersurfaces whose support functions are of class C^2. For n = 2, the problem is already very difficult: if R in C(S^2;R) changes sign on S^2, the question of existence, uniqueness and regularity of a hedgehog of which R is the curvature function boils down to the study of a MongeAmpère equation of mixed type, a class of equations for which there is no global result but only local ones by Lin and Zuily. First partial studies of this problem have been given in :


