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Local solvability of Cauchy’s problem in the space of complex analytic branching functions. (English. Russian original) Zbl 0797.35002
Differ. Equations 28, No. 12, 1755-1765 (1992); translation from Differ. Uravn. 28, No. 12, 2110-2123 (1992).
Let \(\widehat H (x,\partial/ \partial x) = \sum_{| \alpha | \leq m} a_ \alpha (x)\) \((\partial/ \partial x)^ \alpha\) be a differential operator of order \(m\), where \(x \in \mathbb{C}^ n\), \(a_ \alpha\) are holomorphic functions in a neighborhood \(U\) of a point \(x_ 0\), \(X = \{s(x)= 0)\}\) an analytic submanifold in \(U\) containing \(x_ 0\) and such that \(\text{codim} X = 1\), \(H(x,\partial/ \partial x) = \sum_{| \alpha | = m} a_ \alpha(x)\) \((\partial/ \partial x)^ \alpha\), \(H(x,\partial s/ \partial x) |_ X \not \equiv 0\). Let (1) \(\widehat H (x,\partial/ \partial x) u(x) = f(x)\) in \(U\), \(u(x) \equiv 0 \pmod m\) on \(X\). Using a notion of elementary solution the authors prove the following theorem:
If \(f:U \to \mathbb{C}\) is a multivalued analytic function with singularities in an analytic subset \(Y\), and \(Y\) does not contain \(X\) as an irreducible component, then the Cauchy problem (1) has only one solution in a neighborhood of \(x_ 0\). Singularities of this solution lie in the sum of \(Y\) and the characteristic conoid, cone\((X \cup Y)\), of the sum \(X \cup Y\).
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35A08 Fundamental solutions to PDEs
35G10 Initial value problems for linear higher-order PDEs
35A20 Analyticity in context of PDEs