Holomorphic approximation and estimates for the \({\bar \partial}\)-equation on strictly pseudoconvex nonsmooth domains
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.
non-smooth strictly pseudoconvex domains
nonisotropic regularization of holomorphic functions
scientific article; zbMATH DE number 4051999
scientific article