Related papers: A comparison principle for parabolic complex Monge…
In this paper, we study weak solutions to complex Monge-Amp\`ere equations of the form $(\omega + dd^c \varphi)^n= F(\varphi,.)d\mu$ on a bounded strictly pseudoconvex domain in $\mathbb{C}^n$, where $\omega$ is a smooth $(1,1)$-form,…
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 study the asymptotics of complete Kaehler-Einstein metrics on strictly pseudoconvex domains in C^n and derive a convergence theorem for solutions to the corresponding Monge-Ampere equation. If only a portion of the boundary is analytic,…
The aim of this paper is to give a new proof of the complete characterization of measures for which there exist a solution of the Dirichlet problem for the complex Monge-Ampere operator in the set of plurisubharmonic functions with finite…
The theory of viscosity solutions has been effective for representing and approximating weak solutions to fully nonlinear Partial Differential Equations (PDEs) such as the elliptic Monge-Amp\`ere equation. The approximation theory of…
Let $\Omega\subseteq M$ be a bounded domain with a smooth boundary $\partial\Omega$, where $(M,J,g)$ is a compact, almost Hermitian manifold. The main result of this paper is to consider the Dirichlet problem for a complex Monge-Amp\`{e}re…
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.
In this note, we prove a uniqueness result, up to a positive multiplicative constant, for nontrivial convex solutions to a system of Monge-Amp\`ere equations \begin{equation*} \left\{ \begin{alignedat}{2} \det D^2 u~& = \gamma…
We study regularity properties of solutions to the Dirichlet problem for the complex Homogeneous Monge-Amp\`ere equation. We show that for certain boundary data on $\mathbb P^1$ the solution $\Phi$ to this Dirichlet problem is connected via…
We study the parabolic flow for generalized complex Monge-Amp\`ere type equations on closed Hermitian manifolds. We derive {\em a priori} $C^\infty$ estimates for normalized solutions, and then prove the $C^\infty$ convergence.
We make a systematic study of (quasi-)plurisubharmonic envelopes on compact K\"ahler manifolds, as well as on domains of $\mathbb{C}^n$, by using and extending an approximation process due to Berman [Ber13]. We show that the quasi-psh…
We prove the smoothness of weak solutions to an elliptic complex Monge-Ampere equation, using the smoothing property of the corresponding parabolic flow.
We study the long-time existence and convergence of general parabolic complex Monge-Ampere type equations whose second order operator is not necessarily convex or concave in the Hessian matrix of the unknown solution.
We shall consider the regularity problem of solutions for complex Monge-Ampere equations. First we prove interior $C^2$ estimates of solutions in a bounded domain for complex Monge-Ampere equation with assumption of certain $L^p$ bound for…
The present paper provides two necessary and sufficient conditions for the existence of solutions to the exterior Dirichlet problem of the Monge-Amp\`ere equation with prescribed asymptotic behavior at infinity. By an adapted smooth…
We introduce a parabolic analogue of the elliptic split-type Monge-Amp\`ere equation developed by Fang and the author, extending Streets' twisted Monge-Amp\`ere equation. The resulting equation is fully nonlinear and non-concave. We prove…
We prove the existence and uniqueness of weak solutions for the generalized Monge-Amp\`ere equation and the supercritical deformed Hermitian-Yang-Mills equation in cohomology classes lying on the boundary of the solvable region. Moreover,…
We study the equation $\dot{u}=\log\det (u_{\alpha\bar{\beta}})+f(t,z,u)$ in domains of $\mathbb C^n$. This equation has a close connection with the K\"ahler-Ricci flow. In this paper, we consider the case of the boundary conditions are…
We apply the Monte Carlo method to solving the Dirichlet problem of linear parabolic equations with fractional Laplacian. This method exploit- s the idea of weak approximation of related stochastic differential equations driven by the…
We study H\"older continuity of solutions to the Dirichlet problem for measures having density in $L^p$, $p>1$, with respect to Hausdorff-Riesz measures of order $2n-2+\epsilon$ for $0<\epsilon \leq 2$, in a bounded strongly hyperconvex…