On the Cauchy problem for the Laplace equation. (Russian) Zbl 0773.31007

Shlapunov, A. A.

Sib. Mat. Zh. 33, No. 3, 205-215 (1992).

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

Reviewer: M.Brzezina (Ostrava)

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

Keywords:

necessary and sufficient conditions; harmonic continuation