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The Schwarz potential and singularities of solutions to the branching Cauchy problem. (English. Russian original) Zbl 0881.31006
Dokl. Math. 51, No. 2, 215-217 (1995); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 341, No. 3, 307-309 (1995).
Let $$L$$ be an elliptic operator with constant coefficients in its principal part and other coefficients be real entire functions of finite order on the space $$\mathbb{C}^n$$, and let $$\Omega$$ be a domain with algebraic boundary $$\Gamma$$. The Schwarz potential of the domain $$\Omega$$ relative to the operator $$L$$ is defined as the solution $$W(x)$$ of the special Cauchy problem $$LW=1$$, $$W$$ has a zero of order $$m$$ on the complexification $$\Gamma_C$$ of $$\Gamma$$, where $$m$$ is the order of $$L$$. Under additional conditions on $$\Gamma$$ the authors prove that the set of singularities of the solution to the analytic Cauchy problem $$Lu=f$$, $$u$$ has a zero of order $$m$$ on $$\Gamma_C$$, for any entire function $$f$$ is a subset of singularities of the Schwarz potential.
The case $$n=2$$ was proved by I. N. Vekua earlier. If $$L=\Delta$$ then the proved theorem was conjectured by H. Shapiro. D. Khavinson (1991) proved that the theorem is not true for arbitrary operators with variable coefficients.
##### MSC:
 31B15 Potentials and capacities, extremal length and related notions in higher dimensions 35J30 Higher-order elliptic equations