Solvability of generalized div-curl system and Friedrichs inequalities

In this article, we consider the solvability of two generalized div-curl systems. They are referred to as the equations of magnetostatics and electro- , resp.. Necessary compatibility conditions on the data for the existence of solutions are fully described, and existence of $W^{m, p}$-solutions is proved. Moreover, we give some description for the null spaces. As a corollary, we give the estimates of gradient via generalized ${\rm div}$ and ${\rm curl}$ in $L^p$-framework, which can be considered as Friedrichs inequalities. Furthermore, two decompositions of Sobolev spaces are given.