Symplectic structures and symmetries of solutions of the complex Monge-Ampére equation

Nagoya Mathematical Journal ◽

10.1017/s0027763000025058 ◽

1998 ◽

Vol 150 ◽

pp. 63-83

Author(s):

Stanley M. Einstein-Matthews

Keyword(s):

Dirichlet Problem ◽

Cotangent Bundle ◽

Symplectic Structure ◽

Symplectic Structures ◽

Absolutely Continuous ◽

Nonlinear Dirichlet Problem ◽

Homogeneous Complex ◽

Regularity Results ◽

Monge Ampere

Abstract.The graphs that arise from the gradients of solutions u of the homogeneous complex Monge-Ampère equation are characterized in terms of the natural symplectic structure on the cotangent bundle. This characterization is invariant under symplectic biholomorphisms. Using the symplectic structures we construct symmetries (to be called Lempert transformations) for real valued functions u which are absolutely continuous on lines. We then use these symmetries to generate interesting solutions to the homogeneous complex Monge-Ampère equation and to transform the Poincaré-Lelong equation and the ∂-equation. An example of Lempert transform is given and the main theorem is applied to prove regularity results for exterior nonlinear Dirichlet problem for the homogeneous complex Monge-Ampère equation.

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Refined Boundary Behavior of the Unique Convex Solution to a Singular Dirichlet Problem for the Monge–Ampère Equation

Advanced Nonlinear Studies ◽

10.1515/ans-2017-6045 ◽

2018 ◽

Vol 18 (2) ◽

pp. 289-302

Author(s):

Zhijun Zhang

Keyword(s):

Dirichlet Problem ◽

Crucial Role ◽

Blow Up ◽

Boundary Behavior ◽

Structure Condition ◽

Bounded Smooth Domain ◽

Convex Solution ◽

Bounded Smooth ◽

Monge Ampere ◽

Singular Dirichlet Problem

AbstractThis paper is concerned with the boundary behavior of the unique convex solution to a singular Dirichlet problem for the Monge–Ampère equation\operatorname{det}D^{2}u=b(x)g(-u),\quad u<0,\,x\in\Omega,\qquad u|_{\partial% \Omega}=0,where Ω is a strictly convex and bounded smooth domain in{\mathbb{R}^{N}}, with{N\geq 2},{g\in C^{1}((0,\infty),(0,\infty))}is decreasing in{(0,\infty)}and satisfies{\lim_{s\rightarrow 0^{+}}g(s)=\infty}, and{b\in C^{\infty}(\Omega)}is positive in Ω, but may vanish or blow up on the boundary. We find a new structure condition ongwhich plays a crucial role in the boundary behavior of such solution.

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