Boundary conditions on curves for the three-dimensional Laplace operator

Abstract

Boundary conditions on a curve for the three-dimensional Laplace operator are considered in the paper. The result is obtained in terms of a self-adjoint extension of a certain symmetric operator in L₂(R³) and leads to the following formula for the desired boundary condition:$u - \rho \left( {\ell n_\rho + H(z)} \right)\frac{{\partial u}}{{\partial \rho }} \to 0$ as$\rho \to 0$ where ρ is the distance to the curve, and H(z) is a certain real function on this curve.