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The Neumann operator on strongly pseudoconcave domains and other applications of the integral formula method. (Der Neumannoperator auf streng pseudokonkaven Gebieten und andere Anwendungen der Integralformelmethode.) (German) Zbl 0860.32008

Bonner Mathematische Schriften. 238. Bonn: Univ. Bonn, 146 S. (1992).
This work is an extensive study of the Neumann problem for the \(\overline \partial\) operator for functions of several complex variables. It extends results obtained by other authors for strictly pseudo-convex domains to strictly pseudo-concave domains, which are defined as follows: A set \(D\), which is compactly embedded in a subset \(X\) of \(\mathbb{C}^n\) is strictly pseudo-concave, if either
a.) \(X\) is compact and there is a strictly pseudo-convex domain \(D_1\) with \(C^\infty\) boundary, such that \(D=X\setminus \overline{D}_1\),
b.) or there are two strictly pseudo-convex domains, \(D_1\), \(D_2\) with \(D_1\subset\subset D_2 \subset \subset X\) and with \(C^\infty\) boundary, such that \(D=D_2\setminus \overline{D}_1\).
In the introduction a good survey of the known results is given. To study the Neumann operator, the integral formula approach is used.

MSC:

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
47F05 General theory of partial differential operators
32T99 Pseudoconvex domains
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