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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.
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
35J30 Higher-order elliptic equations