In this paper, we establish boundary Hölder gradient estimates for solutions to the linearized Monge-Ampère equations with Lp (n < p ≤ ∞) right hand side and C1,γ boundary values under natural assumptions on the domain, boundary data and the MongeAmpère measure. These estimates extend our previous boundary regularity results for solutions to the linearized Monge-Ampère equations with bounded ri...

The space of Monge-Ampère functions, introduced by J. H. G. Fu in [7, 8] is a space of rather rough functions in which the map u 7→ Det D is well-defined and weakly continuous with respect to a natural notion of weak convergence. We prove a rigidity theorem for Lagrangian integral currents that allows us to extend the original definition of Monge-Ampère functions given in [7]. We also prove tha...

‎this paper presents two main results that the singular values of the hadamard product of normal matrices $a_i$ are weakly log-majorized by the singular values of the hadamard product of $|a_{i}|$ and the singular values of the sum of normal matrices $a_i$ are weakly log-majorized by the singular values of the sum of $|a_{i}|$‎. ‎some applications to these inequalities are also given‎. ‎in addi...

The general solution to the Complex Monge-Ampère equation in a two dimensional space is constructed.

A general solution to the Complex Monge-Ampère equation in a space of arbitrary dimensions is constructed.

We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampère type. These two problems are: the local isometric embedding problem for two-dimensional Riemannian manifolds, and the problem of locally prescribed Gaussian curvature for surfaces in R3 . We prove a general local existence result for a large class of Monge-Amp...

Dispersive deformations of the Monge equation uu = uux are studied using ideas originating from topological quantum field theory and the deformation quantization programme. It is shown that, to a high-order, the symmetries of the Monge equation may also be appropriately deformed, and that, if they exist at all orders, they are uniquely determined by the original deformation. This leads to eithe...