Solution of the transmission problem
The solution of the following transmission problem for the Laplace equation is constructed: \(\Delta u_{+} =0 \) on \( G^{+}\), \( \Delta u_{-} =0 \) in \(G^-\), \( u_{+}-u_{-} =f \) in \(\partial G^{+}\), \(n \cdot (\nabla u_+ - a \nabla u_{-})+b \tau \cdot (\nabla u_{+} - \nabla u_{-})+h_+ u_+ + h_-u_- = g\) on \(\partial G^+\), where \(G^+ \) and \(G^- \) are complementary domains with connected Lipschitz boundary, \(a \) is a positive constant, \(h^+ \) and \(h^- \) are nonnegative functions belonging to a suitable Lebesgue space on \(\partial G^+ \), as \( f \) and \( g \), and \(\tau \) is a measurable field of unit tangent vectors on \(\partial G^+.\)
scientific article; zbMATH DE number 5709158