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On the Cauchy problem for the Laplace equation. (Russian) Zbl 0773.31007
Let $$D\subset \mathbb{R}^ n$$, $$n\geq 2$$, be a bounded connected open set and let $$S\subset D$$ be a closed smooth manifold dividing $$D$$ into two connected sets $$D^ +$$ and $$D^ -$$. The author finds necessary and sufficient conditions in terms of the harmonic continuation of some potentials such that the following problem is solvable: for a given $$f_ 0\in {\mathcal C}^ 1(S)$$ and $$f_ 1\in {\mathcal C}(S)$$ find $$f\in{\mathcal C}^ 1(D^ -\cup S)$$ such that $$f$$ is harmonic on $$D^ +$$ and $$f|_ S=f_ 0$$, $${{\partial f}\over {\partial n}}|_ S=f_ 1$$.

MSC:
 31B20 Boundary value and inverse problems for harmonic functions in higher dimensions 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35J25 Boundary value problems for second-order elliptic equations