×

zbMATH — the first resource for mathematics

On the global solvability of the Cauchy problem in spaces of ramified analytic functions. (English. Russian original) Zbl 0806.35002
Sov. Math., Dokl. 44, No. 3, 836-838 (1992); translation from Dokl. Akad. Nauk SSSR 321, No. 5, 910-913 (1991).
The Cauchy problem for a differential operator \(\widehat H = H(x, \partial /\partial x)\) of order \(m\) with entire coefficients in \(\mathbb{C}^ n\) and Cauchy data on an irreducible analytic set \(X\) of codimension 1 is studied under the assumption that the highest symbol \(H_ m (p)\) of \(\widehat H\) is independent of \(x\) and that the set \(X\) is not everywhere characteristic relative to \(H_ m(p)\). As the main result the authors give an expression for a solution of the Cauchy problem in terms of an elementary solution of the operator \(\widehat H\). This solution is defined in the whole space \(\mathbb{C}^ n\) as a multivalued analytic function with singularities on some analytic set in \(\mathbb{C}^ n\), which can be effectively computed.
Reviewer: J.Fuka (Praha)
MSC:
35A08 Fundamental solutions to PDEs
35C15 Integral representations of solutions to PDEs
58J47 Propagation of singularities; initial value problems on manifolds
32C30 Integration on analytic sets and spaces, currents
PDF BibTeX XML Cite