$L^2$-well-posedness of 3d div-curl boundary value problems

Criteria for the existence and uniqueness of weak solutions of div-curl boundary-value problems on bounded regions in space with $C^2$-boundaries are developed. The boundary conditions are either given normal components of the field or else given tangential components of the field. Under natural integrability assumptions on the data, finite-energy ($L^2$) solutions exist if and only if certain compatibility conditions hold on the data. When compatibility holds, the dimension of the solution space of the boundary-value problem depends on the differential topology of the region. The problem is well-posed with a unique solution in $L^2(\Omega ;\mathbb {R}^3)$ provided, in addition, certain line or surface integrals of the field are prescribed. Such extra integrals are described. These results depend on certain weighted orthogonal decompositions of $L^2$ vector fields which generalize the Hodge-Weyl decomposition. They involve special scalar and vector potentials. The choices described here enable a decoupling of the equations and a weak interpretation of the boundary conditions. The existence of solutions for the equations for the potentials is obtained from variational principles. In each case, necessary conditions for solvability are described and then these conditions are shown to also be sufficient. Finally $L^2$-estimates of the solutions in terms of the data are obtained. The equations and boundary conditions treated here arise in the analysis of Maxwell’s equations and in fluid mechanical problems.