Cegrell, Urban; Persson, Leif The Dirichlet problem for the complex Monge-Ampère operator: Stability in \(L^ 2\). (English) Zbl 0799.32013 Mich. Math. J. 39, No. 1, 145-151 (1992). In a bounded, strictly pseudoconvex domain \(\Omega \subset \mathbb{C}^ n\), consider the Dirichlet problem for the generalized complex Monge-Ampère operator: namely, to find a bounded plurisubharmonic function \(\varphi\) such that \((dd^ c \varphi)^ n = g dV\) in \(\Omega\) and \(\lim_{z \to \xi} \varphi (z) = h(\xi)\) for all \(\xi \in \partial \Omega\).The authors prove that if \(0 \leq g \in L^ 2(\Omega)\) and \(h \in C(\partial \Omega)\), then there is a unique solution \(\varphi = U(h,g)\), and it is continuous not only at the boundary but also in \(\overline \Omega\). Their main result is the stability estimate \[ \sup_ \Omega \bigl | U(h_ 1, g_ 1) - U(h_ 2,g_ 2) \bigr | \leq \sup_{\partial \Omega} | h_ 2-h_ 2 | + C \left( \int_ \Omega | g_ 1 - g_ 2 |^ 2 \right)^{1/2n}, \] which generalizes an inequality of B. Gaveau [J. Funct. Anal. 25, 391-411 (1977; Zbl 0356.35071)], who used the \(L^ \infty\) norm instead of \(L^ 2\). The proof uses a comparison between convex and plurisubharmonic functions and their real and complex Monge-Ampère measures. Reviewer: H.P.Boas (College Station) Cited in 1 ReviewCited in 22 Documents MSC: 32U05 Plurisubharmonic functions and generalizations 35B35 Stability in context of PDEs Keywords:Dirichlet problem; complex Monge-Ampère operator Citations:Zbl 0356.35071 PDFBibTeX XMLCite \textit{U. Cegrell} and \textit{L. Persson}, Mich. Math. J. 39, No. 1, 145--151 (1992; Zbl 0799.32013) Full Text: DOI