The 3-dimensional div-curl system is an overdetermined system of linear partial differential equations that arises in a number of different areas. This talk will describe some recent results on necessary and sufficient conditions for the finite-energy (L² -) solvability of boundary value problems for the system. While the problem may be formulated as a linear problem in a Hilbert space, the description of solutions has required a surprising mixture of topics. They include issues about orthogonal decompositions of L² or H¹ vector fields using subspaces generated by scalar and vector potentials and the use of variational principles to prove both existence and non-existences of solutions of linear equations.

These systems arise from the time-independent Maxwell equations for an electromagnetic field and also in fluid mechanics. Different physical problems impose different boundary conditions and lead to different solvability conditions. These conditions require not only boundary data, but also the specification of certain integrals associated with the differential topology of the region and the boundary data. These extra conditions have important physical interpretations. An open conjecture about the geometrical interpretation of the dimension of the null space associated with certain configurations will be described.