This paper generalizes results of Lempert and Szöke on the structure of the singular set of a solution of the homogeneous Monge-Ampère equation on a Stein manifold. Their a priori assumption that the singular set has maximum dimension is shown to be a consequence of regularity of the solution. In addition, their requirement that the square of the solution be smooth everywhere is replaced by a s...

In this paper we characterize viscosity solutions to nonlinear parabolic equations (including Monge-Ampere equations) by asymptotic mean value formulas. Our formulas can be interpreted from a probabilistic point of view in terms Dynamic Programming Principles for certain two-player, zero-sum games.

0 < λ ≤ f ≤ Λ in Ω, and for some x ∈ Ω, Sh(x) ⊂⊂ Ω, then Sh(x) is equivalent to an ellipsoid centered at x i.e. kE ⊂ Sh(x)− x ⊂ k−1E for some ellipsoid E of volume h and for a constant k > 0 which depends only on λ,Λ, n. This property provides compactness of sections modulo affine transformations. This is particularly useful when dealing with interior C and W 2,p estimates of strictly convex so...

X (F ωM ) , 2 we have the following: (1) (Apriori estimate) Suppose u is a weak solution in PSHF∗ωM (X) ∩ L(X) of the equation with the normalization supX u = 0, then there is a constant C such that ‖u‖L∞ ≤ C‖f‖Lp where C only depend on F , ω and p; (2) There would always be a bounded solution for this equation; (3) If F is locally birational, then any bounded solution is actually the unique co...