We derive gradient and second order a priori estimates for solutions of the Neumann problem for a general class of fully nonlinear elliptic equations on compact Riemannian manifolds with boundary. These estimates yield regularity and existence results.Mathematical Subject Classification (2010): 35B45, 35J15, 35J25, 35J66, 58J05.

No abstract

No abstract

We study the Dirichlet problem for complex Monge-Ampère equations in Hermitian manifolds with general (non-pseudoconvex) boundary. Our main result (Theorem 1.1) extends the classical theorem of Caffarelli, Kohn, Nirenberg and Spruck [12] in C n . We also consider the equation on compact manifolds without boundary, attempting to generalize Yau's theorems [71] in the Kähler case. As applications of the main result we study some connections between the homogeneous complex Monge-Ampère (HCMA) equation and totally real submanifolds, and a special Dirichlet problem for the HCMA equation related to Donaldson's conjecture [23] on geodesics in the space of Kähler metrics.

No abstract