On permeable potential boundary conditions for the Laplace–Beltrami operator

Under study are the so-called permeable potential boundary conditions for the Laplace–Beltrami operator defined in a domain $\Omega$ on the unit sphere $S$ in $\mathbb R^3$. The permeability of boundary conditions means that a solution to a boundary value problem in $\Omega$ coincides with a solution to the Laplace–Beltrami equation on the whole sphere in absence of any boundary conditions.