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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
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