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The Dirichlet problem for the complex Monge-Ampère operator: Stability in \(L^ 2\). (English) Zbl 0799.32013

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.

MSC:

32U05 Plurisubharmonic functions and generalizations
35B35 Stability in context of PDEs

Citations:

Zbl 0356.35071
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