Div-curl boundary value problems model time-independent solutions of Maxwell’s equations and also arise in fluid mechanics. They constitute a linear over-determined system of equations.  In this talk I will describe

1. solvability criteria for the existence of finite energy (L^2-)solutions,
2. the representation of solutions using scalar and vector potentials, and
3. the possible  non-uniqueness of solutions; depending on the  topology of the domain.

These results cover cases where either the normal, or tangential, component of the field is prescribed on the boundary. Also the case of mixed normal and tangential boundary data. The existence results are obtained using variational methods. For each of these cases, the uniqueness of the solutions depends on the differential topology of the domain; there is non-uniqueness when the domain is topologically non-trivial.  To describe a well-posed problem extra integrals of the solution must also be specified – and these extra conditions have  physical interpretations.

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