Finite-energy solutions of mixed 3d div-curl systems

This paper describes the existence and representation of certain finite energy $(L^2$-)solutions of weighted div-curl systems on bounded 3D regions with $C^2$-boundaries and mixed boundary data. Necessary compatibility conditions on the data for the existence of solutions are described. Subject to natural integrability assumptions on the data, it is then shown that there exist $L^2$-solutions whenever these compatibility conditions hold. The existence results are proved by using a weighted orthogonal decomposition theorem for $L^2$-vector fields in terms of scalar and vector potentials. This representation theorem generalizes the classical Hodge-Weyl decomposition. With this special choice of the potentials, the mixed div-curl problem decouples into separate problems for the scalar and vector potentials. Variational principles for the solutions of these problems are described. Existence theorems, and some estimates, for the solutions of these variational principles are obtained. The unique solution of the mixed system that is orthogonal to the null space of the problem is found and the space of all solutions is described.

The second part of the paper treats issues concerning the non-uniqueness of solutions of this problem. Under additional assumptions, this space is shown to be finite dimensional and a lower bound on the dimension is described. Criteria that prescribe the harmonic component of the solution are investigated. Extra conditions that determine a well-posed problem for this system on a simply connected region are given. A number of conjectures regarding the results for bounded regions with handles are stated.