Complex-analytic Cauchy problem in a bounded domain.

*(English. Russian original)*Zbl 0704.35026
Differ. Equations 26, No. 1, 121-131 (1990); translation from Differ. Uravn. 26, No. 1, 136-147 (1990).

This article deals with the global solvability of linear partial differential equation \(H\hat u=f\) of order m with constant coefficients in the space of (multi-valued) holomorphic functions on a domain \(D\subset {\mathbb{C}}^ n\) with (branching) singularity along a complex hypersurface \(X=\{s(x)=0\}\). Let \(A_ q(X)\) denote the space of such functions satisfying the estimate \(| f| \leq C| s(x)|^ q\). D is called \((H,X)\)-convex if every bicharacteristic line starting from a point of \(X\setminus \partial D\) with the holomorphic conormal direction to \(\partial D\), or from a point of \(X\cap \partial D\) with a holomorphic conormal direction to \(X\cap \partial D\). Then if X has no characteristic points in Int(D), \(\hat Hu=f\in A_ q(X)\) is solvable in \(u\in A_{q+m}(X)\). If there are some, it is solvable in the space of functions having singularity further along the subvariety Y formed by the bicharacteristic lines emanating from these characteristic points. The proof is made on examination of the singularity of the solution given by the Radon integral.

Reviewer: A.Kaneko

##### MSC:

35E20 | General theory of PDEs and systems of PDEs with constant coefficients |

35C15 | Integral representations of solutions to PDEs |

35A20 | Analyticity in context of PDEs |