# zbMATH — the first resource for mathematics

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