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Dirichlet problem for degenerate elliptic complex Monge-Ampère equation. (English) Zbl 1217.35088

Summary: We consider the Dirichlet problem \[ \det \biggl({\frac{\partial^2u}{\partial z_i\partial \overline{z_j}}} \biggr)=g(z,u)\quad\text{in }\Omega, \quad u\big|_{ \partial \Omega}=\varphi, \] where \(\Omega\) is a bounded open set of \(\mathbb{C}^n\) with regular boundary, \(g\) and \(\varphi\) are sufficiently smooth functions, and \(g\) is non-negative. We prove that, under additional hypotheses on \(g\) and \(\varphi\), if \(|\det \varphi _{i\overline{j}}-g|_{C^{s_{\ast}}}\) is sufficiently small the problem has a plurisubharmonic solution.
Editorial remark: There are no references given to the corresponding results for the real Monge-Ampère equation in dimension two [see K. Amano, Bull. Aust. Math. Soc. 37, No. 3, 389–410 (1988), corrigendum 38, No. 3, 479–480 (1988; Zbl 0652.35032)] and in dimension \(n\) [see A. Atallah, Trans. Am. Math. Soc. 352, No. 6, 2701–2721 (2000; Zbl 0971.35019)], however, the technical methods are exactly the same as in Atallah’s paper.

MSC:

35J96 Monge-Ampère equations
32W20 Complex Monge-Ampère operators
35J70 Degenerate elliptic equations
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