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Bases with double orthogonality in the Cauchy problem for systems with injective symbols. (English) Zbl 0828.35040
The authors consider the Cauchy problem for solutions of an equation $$Pf = 0$$ where $$P$$ is a differential operator with injective symbol on an open set $$X \subset \mathbb{R}^n$$. The operator $$P$$ is given by an $$(l \times k)$$-matrix of scalar differential operators whose orders are equal or less than $$p$$ on $$X$$. The most important class of operators with injective symbols is the class of elliptic differential operators corresponding to the case $$l = k$$. The model example of other type systems is the Cauchy-Riemann system in the space $$\mathbb{C}^n$$ $$(n > 1)$$. The authors obtain solvability condition and a Carleman formula for the solution of the problem; the constructed bases with double orthogonality and the trace theorems for the solutions of the equation $$Pf = 0$$ on the smooth boundary of the domain are essentially used. The paper contains a detailed survey of results in the considered subjects. The authors consider also examples of systems of simplest type.

MSC:
 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.) 32A40 Boundary behavior of holomorphic functions of several complex variables 32D15 Continuation of analytic objects in several complex variables
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