Plurisubharmonic defining functions of weakly pseudoconvex domains in \({\mathbb{C}}^ 2\) (Q790305)

Bounded pseudoconvex domains \(\Omega_ 1\) and \(\Omega_ 2\) in \({\mathbb{C}}^ 2\) with real-analytic smooth boundaries, \(0\in b\Omega_ i\), \(i=1,2\), with the following properties are constructed: For \(i=1,2\) let \(U_ i=U_ i(0)\) be open neighborhoods of the origin. Then there exist no functions \(\sigma_ i:U_ i\to {\mathbb{R}}\) such that \(U_ i\cap \Omega_ i=\{z\in {\mathbb{C}}^ 2| \sigma_ i(z)<0\},\) the gradient \(d\sigma_ i\neq 0\) on \(b\Omega_ i\cap U_ i\) and the Hermitian form given by the matrix \((a_{jk})\) with \(a_{jk}=\partial^ 2\sigma_ i/\partial z_ j\partial \bar z_ k\) is positive semidefinite in all points \(p\in b\Omega_ i\cap U_ i\) on \({\mathbb{C}}^ 2.\) Furthermore the origin is the only non-strongly pseudoconvex boundary point of \(b\Omega_ 1\) and \(\Omega_ 1\) has a local holomorphic supporting function in the origin. The set of the non-strongly pseudoconvex boundary-points of \(\Omega_ 2\) is a smooth real-analytic curve through the origin.