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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 \(X \cup Y\).
MSC:
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
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