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Let $(X, omega)$ be a compact Kahler manifold of complex dimension n and $theta$ be a smooth closed real $(1,1)$-form on $X$ such that its cohomology class ${ theta }in H^{1,1}(X, mathbb{R})$ is pseudoeffective. Let $varphi$ be a $theta$-psh function, and let $f$ be a continuous function on $X$ with bounded distributional laplacian with respect to $omega$ such that $varphi leq f. $ Then the non-pluripolar measure $theta_varphi^n:= (theta + dd^c varphi)^n$ satisfies the equality: $$ {bf{1}}_{{ varphi = f }} theta_varphi^n = {bf{1}}_{{ varphi = f }} theta_f^n,$$ where, for a subset $Tsubseteq X$, ${bf{1}}_T$ is the characteristic function. In particular we prove that [ theta_{P_{theta}(f)}^n= { bf {1}}_{{P_{theta}(f) = f}} theta_f^nqquad {rm and }qquad theta_{P_theta[varphi](f)}^n = { bf {1}}_{{P_theta[varphi](f) = f }} theta_f^n. ]
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We review recent advances in the numerical analysis of the Monge-Amp`ere equation. Various computational techniques are discussed including wide-stencil finite difference schemes, two-scaled methods, finite element methods, and methods based on geometric considerations. Particular focus is the development of appropriate stability and consistency estimates which lead to rates of convergence of the discrete approximations. Finally we present numerical experiments which highlight each method for a variety of test problem with different levels of regularity.
We study complex geodesics and complex Monge-Amp`{e}re equations on bounded strongly linearly convex domains in $mathbb C^n$. More specifically, we prove the uniqueness of complex geodesics with prescribed boundary value and direction in such a domain, when its boundary is of minimal regularity. The existence of such complex geodesics was proved by the first author in the early 1990s, but the uniqueness was left open. Based on the existence and the uniqueness proved here, as well as other previously obtained results, we solve a homogeneous complex Monge-Amp`{e}re equation with prescribed boundary singularity, which was first considered by Bracci et al. on smoothly bounded strongly convex domains in $mathbb C^n$.
In this paper we consider the Monge-Amp`{e}re type equations on compact almost Hermitian manifolds. We derive a priori estimates under the existence of an admissible $mathcal{C}$-subsolution. Finally, we also obtain an existence theorem if there exists an admissible supersolution.
We prove the existence of a continuous quasi-plurisubharmonic solution to the Monge-Amp`ere equation on a compact Hermitian manifold for a very general measre on the right hand side. We admit measures dominated by capacity in a certain manner, in particular, moderate measures studied by Dinh-Nguyen-Sibony. As a consequence, we give a characterization of measures admitting Holder continuous quasi-plurisubharmonic potential, inspired by the work of Dinh-Nguyen.
Let $Omegasubseteq M$ be a bounded domain with smooth boundary $partialOmega$, where $(M,J,g)$ is a compact almost Hermitian manifold. Our main result of this paper is to consider the Dirichlet problem for complex Monge-Amp`{e}re equation on $Omega$. Under the existence of a $C^{2}$-smooth strictly $J$-plurisubharmonic ($J$-psh for short) subsolution, we can solve this Dirichlet problem. Our method is based on the properties of subsolution which have been widely used for fully nonlinear elliptic equations over Hermitian manifolds. %This work was already done by Plis when we assume there is a strictly $J$-psh defining function for $Omega$.
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