Entire Invariant Solutions to Monge-ampère Equations

Boundary Regularity for Solutions to the Linearized Monge-ampère Equations

We obtain boundary Hölder gradient estimates and regularity for solutions to the linearized Monge-Ampère equations under natural assumptions on the domain, Monge-Ampère measures and boundary data. Our results are affine invariant analogues of the boundary Hölder gradient estimates of Krylov.

Lift of Invariant to Non-Invariant Solutions of Complex Monge-Ampère Equations

We show how partner symmetries of the elliptic and hyperbolic complex Monge-Ampère equations (CMA and HCMA) provide a lift of non-invariant solutions of threeand two-dimensional reduced equations, i.e., a lift of invariant solutions of the original CMA and HCMA equations, to non-invariant solutions of the latter four-dimensional equations. The lift is applied to non-invariant solutions of the t...

Global W2, p estimates for solutions to the linearized Monge–Ampère equations

In this paper, we establish global W 2,p estimates for solutions to the linearizedMonge–Ampère equations under natural assumptions on the domain, Monge– Ampère measures and boundary data. Our estimates are affine invariant analogues of the global W 2,p estimates of Winter for fully nonlinear, uniformly elliptic equations, and also linearized counterparts of Savin’s global W 2,p estimates for th...

Existence of convex solutions for systems of Monge-Ampère equations

Convex Solutions of Systems Arising from Monge-ampère Equations

We establish two criteria for the existence of convex solutions to a boundary value problem for weakly coupled systems arising from the Monge-Ampère equations. We shall use fixed point theorems in a cone.