Related papers: Two-scale method for the Monge-Amp\`ere Equation: …
This is mostly an exposition, aimed to be accessible to geometers, analysts, and probabilists, of a fundamental recent theorem of R. Berman with recent developments by J. Hultgren, that asserts that the second boundary value problem for the…
We solve the Dirichlet problem for the complex Monge-Amp\`ere equation on a strictly pseudoconvex with the right hand side being a positive Borel measure which is dominated by the Monge-Amp\`ere measure of a H\"older continuous…
We consider the numerical solution of the optimal transport problem between densities that are supported on sets of unequal dimension. Recent work by McCann and Pass reformulates this problem into a non-local Monge-Amp\`ere type equation.…
This is the content of the lectures given by the author at the winter school KAWA3 held at the University of Barcelona in 2012 from January 30 to February 3. The main goal was to give an account of viscosity techniques and to apply them to…
This article is about the convex solution $u$ of the Monge--Amp\`ere equation on an at least 2-dimensional open bounded convex domain with Dirichlet boundary data and nonnegative bounded right-hand side. For convex functions with zero…
The Monge-Amp\`ere equation is a fully nonlinear partial differential equation (PDE) of fundamental importance in analysis, geometry and in the applied sciences. In this paper we solve the Dirichlet problem associated with the…
In this paper, we are interested in studying the Dirichlet problem for the complex Monge-Amp\`ere operator. We provide necessary and sufficient conditions for the problem to have H\"older continuous solutions.
We investigate global H\"older gradient estimates for solutions to the Monge-Amp\`ere equation $$\mathrm{det}\;D^2 u=f\quad\mathrm{in}\;\Omega,$$ where the right-hand side $f$ is bounded away from $0$ and $\infty$. We consider two main…
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 present two numerical methods for the fully nonlinear elliptic Monge-Ampere equation. The first is a pseudo transient continuation method and the second is a pure pseudo time marching method. The methods are proven to converge to a…
We prove the H\"{o}lder continuity of the unique solution to quaternionic Monge-Amp\`{e}re equation with densities in $L^{p},$ $p>2,$ on a bounded strictly pseudoconvex domains.
In this paper we consider the generalised solutions to the Monge-Amp{\`{e}}re type equations with general source terms. We firstly prove the so-called comparison principle and then give some important propositions for the border of…
In this paper we shall prove the existence, uniqueness and global H$\ddot{o}$lder continuity for the Dirichlet problem of a class of Monge-Amp\`ere type equations which may be degenerate and singular on the boundary of convex domains. We…
We show the existence and uniqueness of bounded solutions to the degenerate complex Monge-Amp\`ere type equations on compact Hermitian manifolds. We also study the asymptotics of these solutions. As applications, we give partial answers to…
This paper is concerned with developing and analyzing convergent semi-Lagrangian methods for the fully nonlinear elliptic Monge-Amp\`ere equation on general triangular grids. This is done by establishing an equivalent (in the viscosity…
We study the regularity of solutions to complex Monge-Amp\`ere equations $(dd^c u)^n=f dV$, on bounded strongly pseudoconvex domains $ \Omega \subset \C^n$. We show, under a mild technical assumption, that the unique solution $u$ to such an…
This paper concerns with the convergence analysis of a fourth order singular perturbation of the Dirichlet Monge-Amp\`ere problem in the $n$-dimensional radial symmetric case. A detailed study of the fourth order problem is presented. In…
In this paper, we study the Cauchy-Dirichlet problem for Parabolic complex Monge-Amp\`ere equations on a strongly pseudoconvex domain by the viscosity method. We extend the results in [EGZ15b] on the existence of solution and the…
In this paper, we establish boundary H\"older gradient estimates for solutions to the linearized Monge-Amp\`ere equations with $L^{p}$ ($n<p\leq\infty$) right hand side and $C^{1,\gamma}$ boundary values under natural assumptions on the…
We study the exact effect of the anisotropic convexity of domains on the boundary estimate for two Monge-Amp\`ere Equations: one is singular which is from the proper affine hyperspheres with constant mean curvature; the other is degenerate…