Related papers: A Dirichlet type problem for non-pluripolar comple…

We construct solutions to Monge-Amp\`ere equations whose Monge-Amp\`ere measures contain singular components supported on low codimensional sets. We also study the regularity of such solutions. To motivate our construction, we present…

A general solution to the Complex Monge-Amp\`ere equation in a space of arbitrary dimensions is constructed.

This paper studies the numerical approximation of solution of the Dirichlet problem for the fully nonlinear Monge-Ampere equation. In this approach, we take the advantage of reformulation the Monge-Ampere problem as an optimization problem,…

By constructing appropriate smooth, possibly non-convex supersolutions, we establish sharp lower bounds near the boundary for the modulus of nontrivial solutions to singular and degenerate Monge-Amp\`ere equations of the form $\det D^2 u…

Let $\Omega$ be a bounded strictly pseudoconvex domain of $\mathbb{C}^n$. We solve degenerate complex Monge-Amp\`ere equations of the form $(\omega + dd^c \varphi)^n = \mu$ in the generalized Cegrell classes $\mathcal{K}(\Omega,\omega,H)$,…

In this article we address the question whether the complex Monge-Amp\`{e}re equation is solvable for measures with large singular part. We prove that under some conditions there are no solution when the right-hand side is carried by a…

Extending a recent theory developed on compact K\"ahler manifolds by Guedj-Lu-Zeriahi and the author, we define and study pluripotential solutions to degenerate parabolic complex Monge-Amp\`ere equations on compact Hermitian manifolds.…

In this paper, we study the Dirichlet problem for the implicit degen- erate nonlinear elliptic equation with variable exponent in a bounded domain. We obtain sufficient conditions for the existence of a solution with- out regularization and…

We present two comparison principles for viscosity sub- and supersolutions of Monge-Ampere-type equations associated to a family of vector fields. In particular, we obtain the uniqueness of a viscosity solution to the Dirichlet problem for…

The present paper is devoted to the study of the Dirichlet problem ${\rm{Re}}\,\omega(z)\to\varphi(\zeta)$ as $z\to\zeta,$ $z\in D,\zeta\in \partial D,$ with continuous boundary data $\varphi :\partial D\to\mathbb R$ for Beltrami equations…

Let $(X,\omega)$ be a compact $n$-dimensional K\"ahler manifold on which the integral of $\omega^n$ is $1$. Let $K$ be an immersed real $\mathcal{C}^3$ submanifold of $X$ such that the tangent space at any point of $K$ is not contained in…

This paper introduces a fast and robust iterative scheme for the elliptic Monge-Amp\`ere equation with Dirichlet boundary conditions. The Monge-Amp\`ere equation is a nonlinear and degenerate equation, with applications in optimal…

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…

In this paper, we obtain gradient estimates and Laplacian estimates for the solution to the singular complex Monge-Amp\`ere equation by applying the integral method.

We prove the existence and uniqueness of non-trivial stable solutions to Landau-Lifshitz-Maxwell equations with Dirichlet boundary condition for large anisotropies and small domains, where the domains are non-simply connected.

Let $\mathcal{M}_1$ denote the space of solutions $z(x,y)$ to an elliptic, real analytic Monge-Amp\`ere equation ${\rm det} (D^2 z)=\varphi(x,y,z,Dz)>0$ whose graphs have a non-removable isolated singularity at the origin. We prove that…

Let $\phi: \Bbb R^n \mapsto \Bbb R$ be a strictly convex and smooth function, and $\mu= \text{det}\,D^2 \phi$ be the Monge-Amp\`ere measure generated by $\phi.$ For $x\in \Bbb R^n$ and $t>0$, let $S(x,t):=\{y\in \Bbb R^n:…

We describe the behavior of the singularities of solutions to degenerate complex Monge-Amp`ere equations on K\"ahler manifolds. This was not resolved since the fundamental paper of S-T Yau \cite{y} on this subject.

We present a numerical method for solving the Monge-Ampere equation based on the characterization of the solution of the Dirichlet problem as the minimizer of a convex functional of the gradient and under convexity and nonlinear…

We consider the numerical approximation of surfaces of prescribed Gaussian curvature via the solution of a fully nonlinear partial differential equation of Monge-Amp\`ere type. These surfaces need not be continuous up to the boundary of the…