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Register a new userComplex geodesics and complex Monge-Amp`{e}re equations with boundary singularity
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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 $mathbb C^n$.
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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. ]
Let $X$ be a compact Kahler manifold of dimension $n$ and $omega$ a Kahler form on $X$. We consider the complex Monge-Amp`ere equation $(dd^c u+omega)^n=mu$, where $mu$ is a given positive measure on $X$ of suitable mass and $u$ is an $omega$-plurisubharmonic function. We show that the equation admits a Holder continuous solution {it if and only if} the measure $mu$, seen as a functional on a complex Sobolev space $W^*(X)$, is Holder continuous. A similar result is also obtained for the complex Monge-Amp`ere equations on domains of $mathbb{C}^n$.
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.
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.
A new proof for stability estimates for the complex Monge-Amp`ere and Hessian equations is given, which does not require pluripotential theory. A major advantage is that the resulting stability estimates are then uniform under general degenerations of the background metric in the case of the Monge-Amp`ere equation, and under degenerations to a big class in the case of Hessian equations.
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