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تسجيل مستخدم جديدThe Monge-Amp`{e}re equation
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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.
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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
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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 $Omega$.
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wever, it applies quite generally -- perhaps most importantly to symplectic manifolds equipped with a Gromov metric. The Lagrange Monge-Ampere operator is an explicit polynomial on ${rm Sym}^2(TX)$ whose principle branch defines the space of Lag-harmonics. Interestingly the operator depends only on the Laplacian and the SKEW-Hermitian part of the Hessian. The Dirichlet problem for this operator is solved in both the homogeneous and inhomogeneous cases. The homogeneous case is also solved for each of the other branches. This paper also introduces and systematically studies the notions of Lagrangian plurisubharmonic and harmonic functions, and Lagrangian convexity. An analogue of the Levi Problem is proved. In ${bf C}^n$ there is another concept, Lag-plurihamonics, which relate in several ways to the harmonics on any domain. Parallels of this Lagrangian potential theory with standard (complex) pluripotential theory are constantly emphasized.
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