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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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