Related papers: Some multi-valued solutions to Monge-Ampere equati…

We show the existence of the strong solutions of the Monge-Ampere equation for the log-concave measures without any regularity assumption. In particular the measures with density of the form $\exp -f$, where $f$ is an $H$-convex function…

In this article, we introduce and study three numerical methods for the Dirichlet Monge Amp\`ere equation in two dimensions. The approaches consist in considering new equivalent problems. The latter are discretized by a wide stencil finite…

In this paper, we solve the Dirichlet problem for Monge-Amp\`ere type equations for $(n-1)$-plurisubharmonic functions on Hermitian manifolds.

In this paper we show some new exact solutions for the generalized modified Degasperis$-$Procesi equation (mDP equation)

We study the continuity of solutions to complex Monge-Ampere equations with prescribed singularities. This generalizes the previous results of DiNezza-Lu and the author. As an application, we can run the Monge-Ampere flow starting at a…

In this paper, we prove the H\"older continuity for solutions to the complex Monge-Amp\`ere equations on non-smooth pseudoconvex domains of plurisubharmonic type ${m}$.

In this paper, we give some precise characterizations of existence of solution to the complex Monge - Amp\`ere equation in the classes $\mathcal E_\chi(\Omega)$ and $\mathcal E_{\chi,loc}(\Omega)$.

We study a Monge-Amp\`ere type equation that interpolates the classical {\sigma_2} -Yamabe equation in conformal geometry and the 2-Hessian equation in dimension 4.

A new relativistic invariant version of nonlinear Maxwell equations is offerred. Some properties of these equations are considered.

We introduce generalized Monge-Amp\`ere capacities and use these to study complex Monge-Amp\`ere equations whose right-hand side is smooth outside a divisor. We prove, in many cases, that there exists a unique normalized solution which is…

We study the solvability and uniqueness for several degenerate Monge--Amp\`ere equations including the Monge--Amp\`ere eigenvalue problem in real Euclidean spaces that involve singular Borel measures. Our approach systematically analyzes…

We show that a positive Borel measure of positive finite total mass, on compact Hermitian manifolds, admits a Holder continuous quasi-plurisubharmonic solution to the Monge-Ampere equation if and only if it is dominated locally by…

In this paper we continue the analysis of the two-scale method for the Monge-Amp\`ere equation for dimension $d \geq 2$ introduced in [10]. We prove continuous dependence of discrete solutions on data that in turn hinges on a discrete…

We show a very general existence theorem to the complex Monge-Amp\`ere type equation on hyperconvex domains.

We prove uniform gradient and diameter estimates for a family of geometric complex Monge-Ampere equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge-Ampere equations. We also prove a…

This is an introduction to a particular class of auxiliary complex Monge-Amp\`ere equations which had been instrumental in $L^\infty$ estimates for fully non-linear equations and various questions in complex geometry. The essential…

The elliptic Monge-Ampere equation is a fully nonlinear Partial Differential Equation which originated in geometric surface theory, and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image…

In this paper, we introduce a family of real Monge-Amp\`ere functionals and study their variational properties. We prove a Sobolev type inequality for these functionals and use this to study the existence and uniqueness of some associated…

We present three alternative derivations of the method of characteristics (MOC) for a second order nonlinear hyperbolic partial differential equation. The MOC gives rise to two mutually coupled systems of ordinary differential equations. As…

In this paper, we study the Neumann problem of Monge-Amp\`ere equations in Semi-space. For two dimensional case, we prove that its viscosity convex solutions must be a quadratic polynomial. When the space dimension $n\geq 3$, we show that…