Well-Posedness for Weak Solutions of Axisymmetric Div-Curl Systems

We study the axisymmetrc div-curl system on bounded volumes of revolution with normal and tangential boundary conditions. This vector system of equations arises in classical field theories. In particular, the electrostatic and magnetostatic axisymmetric Maxwell equations are axisymmetric div-curl systems. The analysis is based on orthogonal decompositions of axisymmetric vector fields.

The characterization of the scalar potentials and stream functions in the orthogonal decompositions leads to the analysis of axisymmetric Laplacian boundary value problems. Axisymmetric Laplacian eigenproblems give rise to natural bases for special gradient and curl subspaces for the orthogonal decompositions, and the eigenvalues appear as best constants in energy estimates for solutions of the axisymmetric Laplacian boundary value problems and in energy estimates for the axisymmetric div-curl system.

The results presented are valid for a general class of bounded C2 volumes of revolution with a nonempty and connected intersection with the axis of symmetry. We allow the domain to contain toroidal holes.