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Holomorphic approximation and estimates for the \({\bar \partial}\)-equation on strictly pseudoconvex nonsmooth domains. (English) Zbl 0645.32009

Denote by X a (non-smooth) compact set in \({\mathbb{C}}^ n\) of the form \(X=\{z:\) r(z)\(\leq 0\}\) where r is a function of class \(C^ 2\), strictly plurisubharmonic in a neighborhood of \(\{\) \(z: r(z)=0\}\) and by H(X) the space of functions holomorphic in some neighborhood of X. For \(m=0,1,..\). and \(o\leq s<1\), we consider the Whitney space \(C^{m,s}(X)\), i.e. the space of jets \(F=(u_{\alpha,\beta})\) where \(\alpha\), \(\beta\) are n- multiindexes with \(| \alpha | +| \beta | \leq m\) and \(u_{\alpha,\beta}\) are continuous functions on X such that the formal Taylor rest \(R_ z^{m-| \alpha | -| \beta |}(u_{\alpha,\beta}(w))\) is \(o(| w-z|^{m+s-| \alpha | -| \beta |})\) for z,w\(\in X\), \(| z-w| \to 0.\)
Main results: a) It is described the closure of H(X) in \(C^{m,s}(X).\)
For \(p\geq 1\), let ary and initial conditions). The approach rests on a monotone iterative method, which yields the existence of maximal and minimal solutions along with upper and lower estimates of the exact solution of the equation. An application of the theory to the study of a nonlinear integral equation occuring in epidemiology is also included.
Reviewer: S.Aizicovici

MSC:

32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
32T99 Pseudoconvex domains
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
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