A Riemann-Hilbert problem for the Moisil–Teodorescu system

In a bounded domain with smooth boundary in $\mathbb{R}^3$ we consider the stationary Maxwell equations for a function $u$ with values in $\mathbb{R}^3$ subject to a nonhomogeneous condition $(u,v)_x = u_0$ on the boundary, where $v$ is a given vector field and $u_0$ 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–Lopatinskij 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.