Partial pluricomplex energy and integrability exponents of plurisubharmonic functions

We first prove a quantitative estimate of the volume of the sublevel sets of a plurisubharmonic function in a hyperconvex domain with boundary values 0 (in a quite general sense) in terms of its Monge–Ampère mass in the domain. Then we deduce a sharp sufficient condition on the Monge–Ampère mass of such a plurisubharmonic function φ for exp(−2φ) to be globally integrable as well as locally integrable.     

The affine Plateau problem

In this paper, we study a second order variational problem for locally convex hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. We prove existence, regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions.

     

A research into the numerical method of Dirichlet's problem of complex Monge-Ampère equation on Cartan-Hartogs domain of the third type

Monge-Ampère equation is a nonlinear equation with high degree, therefore its numerical solution is very important and very difficult. In present paper the numerical method of Dirichlet's problem of Monge-Ampère equation on Cartan-Hartogs domain of the third type is discussed by using the analytic method. Firstly, the Monge-Ampère equation is reduced to the nonlinear ordinary differential equation, then the numerical method of the Dirichlet problem of Monge-Ampère equation becomes the numerical method of two point boundary value problem of the nonlinear ordinary differential equation. Secondly, the solution of the Dirichlet problem is given in explicit formula under the special case, which can be used to check the numerical solution of the Dirichlet problem.     

Pseudo-linear superposition principle for the Monge-Ampère equation based on generated pseudo-operations

Through this paper, we consider generated pseudo-operations of the following form: x⊕y=g⁻¹(g(x)+g(y)), x⊙y=g⁻¹(g(x)g(y)), where g is a continuous generating function. Pseudo-linear superposition principle, i.e., the superposition principle with this type of pseudo-operations in the core, for the Monge-Ampère equation is investigated.     

A note on the contact angle boundary condition for Monge-Ampère equations

We give a simple proof of a result of Xinan Ma concerning a necessary condition for the solvability of the two-dimensional Monge-Ampère equation subject to the contact angle or capillarity boundary condition. Our technique works for more general Monge-Ampère equations in any dimension, and also extends to some other boundary conditions.

     

The complex Monge–Ampère equation with infinite boundary value

In this article we consider the complex Monge–Ampère equation with infinite boundary value in bounded pseudoconvex domains. We prove the existence of strictly plurisubharmonic solution to the problem in convex domains under suitable growth conditions. We also obtain, for general pseudoconvex domains, some nonexistence results which show that these growth conditions are nearly optimal.     

Regularity Properties of the Degenerate Monge-Amp`ere Equations on Compact K¨ahler Manifolds

The author establishes a result concerning the regularity properties of the degenerate complex Monge-Ampère equations on compact Kǎhler manifolds.