A Riemann-Hilbert Problem for the Moisil-Teodorescu System

In a bounded domain with smooth boundary in ?³ we consider the stationary Maxwell equations for a function u with values in ?³ subject to a nonhomogeneous condition (u, v)_x = u₀ on the boundary, where v is a given vector field and u₀ a function on the boundary. We specify this problem within the framework of the Riemann-Hilbert boundary value problems for the Moisil-Teodorescu system. This latter is proved to satisfy the Shapiro-Lopaniskij condition if an only if the vector v is at no point tangent to the boundary. The Riemann-Hilbert problem for the Moisil-Teodorescu system fails to possess an adjoint boundary value problem with respect to the Green formula, which satisfies the Shapiro-Lopatinskij condition. We develop the construction of Green formula to get a proper concept of adjoint boundary value problem.