The Dirichlet problem for the Complex Monge-Ampere equation on a bounded pseudoconvex domain in $\mathbb{C}^n$ was studied in Bedford-Taylor's seminal paper wherein they defined $(dd^{c} u)^n$ for locally bounded plurisubharmonic $u$. But, they don't seem to use it in their Perron-type method, instead using a convex-measure-theoretic construction claiming that the upper envelope is not well-behaved. Now that we know more about psh functions, have people studied the Dirichlet problem without using the measure-theoretic construction of Goffman and Serrin? (and reproved Bedford-Taylor's results)
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2 Answers
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There is a result in the paper
Cegrell, Urban On the Dirichlet problem for the complex Monge-Ampère operator. Math. Z. 185 (1984), no. 2, 247–251.
Theorem. Assume that $\Omega$ is strictly pseudoconvex and that $H(t, z)$ is a measurable, bounded and non-negative function on $(-\infty,, \max h] \times \Omega$, where $h$ is a continuous function on $\partial \Omega$. If $H(t,z)$ is continuous on $(-\infty, \max h]$ for every fixed $z \in \partial \Omega$, then the Dirichlet problem $\varphi \in P(\Omega)$, $(dd^c \varphi)^n =H(\varphi, z)dV$ on $\partial \Omega$, $\lim_{z \to \zeta}\varphi(z) = h(\zeta)$ on $\partial \Omega$ has a solution. (Here, $dV$ denotes the Lebesgue measure and $P(\omega)$ is the class of bounded plurisubharmonic functions in $\Omega$.)
This theorem is a generalization of Theorem A in Bedford and Taylor where $H^{1/n}$ is also required to be convex and increasing in $t$.(These properties are due to the fact that $H$ comes from the Goffman-Serrin operator.)
The proof uses fixed point methods.
Further results of this type were obtained (by Cegrell as well as his students and collaborators), mainly in the setting of hyperconvex domains.
answered Mar 14, 2012 at 18:29
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In the book Degenerate Complex Monge-Ampère Equations by V. Guedj, A. Zeriahi, they circumvent the construction of Goffman and Serrin using ideas from viscosity theory. Let $H_n^+$ denote the set of all semi-positive Hermitian $n$ by $n$ matrices and let $\dot{H}_n^+ \subset H_n^+$ denote the matrices with determinant $n^{-n}$. For $H \in \dot{H}_n^+$ they define
$$ \Delta_H := \sum_{i,j=1}^n h_{i,j}\frac{\partial^2}{\partial z_i \partial \bar{z}_j}$$
and show that for $u \in PSH(\Omega) \cap L^\infty(\Omega)$, $0 \leq f \in C^0(\Omega)$,
$$ (dd^cu)^n \geq f \beta^n \iff \Delta_H u \geq f^{1/n} \text{ for all $H \in \dot{H}_n^+$.} $$
The envelopes are then constructed with regards to the second statement.
