A Quaternionic Treatment of the Inhomogeneous Div-rot System
In this paper we study the inhomogeneous div-rot system (div ~f = g0, rot ~f = ~g) where the datum (g0, ~g) consists of a continuous scalar and a continuous vector field, respectively. We embed the system in its appropriate quaternionic setting, and by using the right inverse of the Moisil–Teodorescu operator, we provide a necessary and sufficient condition for the solvability of the system and we describe its general solution. As a byproduct we obtain an explicit integral expression for the right inverse for the operators div and rot. Finally, we show how the same problem could have been studied using algebraic analysis, and we use this different approach to obtain some additional results.
Colombo, F., Luna-Elizarraras, M.E., Sabadini, I., Shapiro, M., & Struppa, D.C. (2012). A quaternionic treatment of the inhomogenous div-rot system. Moscow Mathematical Journal, 12(1), 37-48.
Independent University (Moscow)