Solving the degenerate complex Monge-Ampère equation with one concentrated singularity. (English) Zbl 0531.35020

Lempert, László

Math. Ann. 263, 515-532 (1983).

The degenerate (complex or real) Monge-Ampère equation has very poor regularity properties. That is if a continuous plurisubharmonic function u satisfies \(\det \partial^ 2u/\partial \bar z_ j\partial z_ k=0\) in a smooth strictly pseudoconvex domain in \({\mathbb{C}}^ n\), and the boundary values of u are smooth, then this does not imply that u is smooth in the domain. The paper under review proposes a modification of the above problem, where good regularity results can be obtained. Let \(D\subset {\mathbb{C}}^ n\) be a strictly convex, analytically bounded domain, \(\phi\) :\(\partial D\to {\mathbb{R}}\) analytic, \(Z\in D\) arbitrary, and c a positive number. For a function u, psh in D, real analytic on \(\bar D\backslash \{Z\}\), consider the problem: \(\det \partial^ 2u/\partial \bar z_ j\partial z_ k=0\) in \(D\backslash \{Z\}\); \(u(z)=c \log | z-Z| +O(1)\) as \(z\to Z\); \(u(z)=\phi(z)\) if \(z\in \partial D\). Theorem: There exists a unique solution u, provided \(c>c_ 0(D,Z,\phi).\)
The proof is based on the corresponding result for \(\phi\equiv 0\), treated in the author’s paper [Bull. Soc. Math. Fr. 109, 427-474 (1981; Zbl 0492.32025)], and the implicit function theorem. - Properties of the solution u are also discussed, such as the nature of the singularity of u in Z; it turns out that \(u(z)-c \log | z-Z|\) can be pulled back to an analytic function on the blowing up of D at Z. [In fact, a slightly weaker result is proved in the paper but the same arguments yield this stronger one, too.]

MSC:

35G20	Nonlinear higher-order PDEs
31C10	Pluriharmonic and plurisubharmonic functions
32U05	Plurisubharmonic functions and generalizations
35B60	Continuation and prolongation of solutions to PDEs

Keywords:

concentrated singularity; regularity of degenerate Monge-Ampère equation; plurisubharmonic function

Citations:

Zbl 0492.32025

References:

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[3]	Bedford, E., Taylor, B.A.: The Dirichlet problem for a complex Monge-Ampère equation. Invent. Math.37, 1-44 (1976) · doi:10.1007/BF01418826
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