The complex-analytic geometry of bicharacteristics and the semi-global existence of holomorphic solutions of linear differential equations -- a bridge between the theory of partial differential equations and the theory of holomorphic functions (Q2743992)

Let \(U\subset \mathbb{C}^n\) be an open neighborhood of some \(a\in \mathbb{C}^n.\) Let \(\varphi(z,\overline z)\) be a strictly plurisubharmonic real analytic function defined on \(U\), with \(\varphi(a,\overline a)=0\) and \(\nabla_z\varphi\) never zero on \(U.\) Let \(\Omega=\{z\in U\); \(\varphi<0\},\) \(S=\{z\in U\); \(\varphi=0\},\) and let \({\mathcal O}^-\) be the sheaf of ``boundary values of holomorphic functions from the side of \(\Omega\)''. Let further \(P(z,D_z)\) be a differential operator with holomorphic coefficients defined on \(U.\) One says that \(Pu=f\) is semi-globally solvable near \(z_0\) in \(S\) (with respect to \(S\) from the side of \(\Omega\)) if \(P:{\mathcal O}^-_{z_0}\rightarrow{\mathcal O}^-_{z_0}\) is surjective. In this paper, the authors describe how the semi-global solvability is induced by the complex-analytic geometry of the bicharacteristic curves of the principal symbol of \(P\).NEWLINENEWLINEFor the entire collection see [Zbl 0969.00052].