We consider the following obstacle problem for Monge-Ampere equation detDu = fχ{u>0} and discuss the regularity of the free boundary ∂{u = 0}. We prove that ∂{u = 0} is C if f is bounded away from 0 and ∞, and it is C if f ≡ 1.

In this thesis, we study degenerate Monge-Ampere equations over projective manifolds. The main degeneration is on the cohomology class which is Kähler in classic cases. Our main results concern the case when this class is semi-ample and big with certain generalization to more general cases. Two kinds of arguments are applied to study this problem. One is maximum principle type of argument. The ...

In this paper, we prove long time existence and convergence results for a class of general curvature flows with Neumann boundary condition. This is the first result for the Neumann boundary problem of non Monge-Ampere type curvature equations. Our method also works for the corresponding elliptic setting.

Using the subgroup structure of the generalized Poincaré group P (1, 4), the symmetry reduction of the five-dimensional wave and Dirac equations and Euler–Lagrange– Born–Infeld, multidimensional Monge–Ampere, eikonal equations to differential equations with a smaller number of independent variables is done. Some classes of exact solutions of the investigated equations are constructed.

The maximally complicated arbitrary-dimensional “maximal” Galileon field equations simplify dramatically for symmetric configurations. Thus, spherical symmetry reduces the equations from the D− to the two-dimensional Monge-Ampere equation, axial symmetry to its cubic extension etc. We can then obtain explicit solutions, such as spherical or axial waves, and relate them to the (known) general, b...

The Monge-Ampere equation, plays a central role in the theory of fully non linear equations. In fact we will like to show how the Monge-Ainpere equation, links in some way the ideas comming from the calculus of variations and those of the theory of fully non linear equations. 2000 Mathematics Subject Classification: 35J15, 35J20, 35J70. When learning complex analysis, it was a remarkable fact t...