zbMATH — the first resource for mathematics

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 \(f|_ S=f_ 0\), \({{\partial f}\over {\partial n}}|_ S=f_ 1\).

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