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 Izv. Akad. Nauk SSSR Ser. Mat., 1984, Volume 48, Issue 5, Pages 1109–1118 (Mi izv1509)

Foliations connected with the Monge–Ampère equation in Hartogs domains

N. G. Kruzhilin

Abstract: Let $D$ be a domain in $\mathbf C^2(z,w)$, and $u$ a solution in $D$ of the equation $(\partial\overline\partial u)^2=0$, where $\partial\overline\partial u\ne0$ in $D$. It is known that, for $u\in C^3(D)$, $D$ is foliated into complex curves on which $u$ is harmonic, and $\partial u/\partial z$ and $\partial u/\partial w$ are holomorphic. We show that if $u=u(|z|,w)$ and $D$ is a complete Hartogs domain with axis of symmetry $z=0$, then such a foliation exists even for $u\in C^2(D)$.
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English version:
Mathematics of the USSR-Izvestiya, 1985, 25:2, 419–427

Bibliographic databases:

UDC: 517.5
MSC: Primary 32C42, 35Q99; Secondary 32A07, 32F05

Citation: N. G. Kruzhilin, “Foliations connected with the Monge–Ampère equation in Hartogs domains”, Izv. Akad. Nauk SSSR Ser. Mat., 48:5 (1984), 1109–1118; Math. USSR-Izv., 25:2 (1985), 419–427

Citation in format AMSBIB
\Bibitem{Kru84} \by N.~G.~Kruzhilin \paper Foliations connected with the Monge--Amp\ere equation in Hartogs domains \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1984 \vol 48 \issue 5 \pages 1109--1118 \mathnet{http://mi.mathnet.ru/izv1509} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=764311} \zmath{https://zbmath.org/?q=an:0579.32028} \transl \jour Math. USSR-Izv. \yr 1985 \vol 25 \issue 2 \pages 419--427 \crossref{https://doi.org/10.1070/IM1985v025n02ABEH001290} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1984AZA7900008} `