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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
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@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
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up vote 4 down vote accepted

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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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 (C2-hypersurfaces 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 Monge-Ampère 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 :

  • Yves Martinez-Maure, Uniqueness results for the Minkowski problem extended to hedgehogs, Central European Journal of Mathematic 10, 2012, 440-450.
  • Yves Martinez-Maure, New notion of index for hedgehogs of R^3 and applications, in:Rigidity and related topics in Geometry, European Journal of Combinatorics, 31 (2010), 1037-1049.
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I apologise for (possibly) misunderstanding your reply, but in my case I require the Monge-Ampere equation to be elliptic. The concavity of the equation (as a function of Hermitian matrices) is under question. –  Vamsi Mar 23 '12 at 22:57
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