On the solution of Dirichlet’s problem of
complex Monge-Amp`ere equation on
Cartan-Hartogs domain of the second type*
1
Weiping YIN
1
Xiaolan YIN
2,3
1. School of Mathematical Sciences, Capital Normal University, Beijing
100037, China. e-mail: wyin@mail.cnu.edu.cn Tel: 86-10-68902600
2. Technology Center of Software Engineering, Institute of Software, Chinese
Academy of Sciences, Beijing 100080, China.
3. Graduate University of Chinese Academy of Sciences, Beijing 100049,
China.
Abstract: Complex Monge-Amp`ere equation is a nonlinear equation with
high degree, therefore to get its solution is very difficult. In present paper how
to get the solution of Dirichlet’s problem of Complex Monge-Amp`ere equation
on the Cartan-Hartogs domain of the second type is discussed by using the
analytic method. Firstly, the complex Monge-Amp`ere equation is reduced to
the nonlinear ordinary differential equation, then the solution of the Dirichlet’s
problem of complex Monge-Amp`ere equation is reduced to the solution of two
point boundary value problem of the nonlinear second-order ordinary differ-
ential equation. Secondly, the solution of the Dirichlet’s problem is given in
semi-explicit formula, and under the special case the exact solution is obtained.
These results may be helpful for the numerical method of Dirichlet’s problem
of complex Monge-Amp`ere equation on the Cartan-Hartogs domain.
Key Words Complex Monge-Amp`ere equation, Dirichlet’s problem,
Cartan-Hartogs domain, Kaehler-Einstein metric.
MR(1991)Subject Classification: 65E05, 32C17, 53C55, 35G30.
1
*This project was supported by the National Natural Science Foundation of China(Grant
No.10771144).
1
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