 
Summary: FINITE ENERGY SOLUTIONS OF MIXED
3D DIVCURL SYSTEMS
GILES AUCHMUTY AND JAMES C. ALEXANDER
Abstract. This paper describes the existence and representation of certain finite energy
(L2
) solutions of weighted divcurl systems on bounded 3d regions with C2
boundaries
and mixed boundary data. Necessary compatibility conditions on the data for the exis
tence of solutions are described. Subject to natural integrability assumptions on the data,
it is then shown that there exist L2
solutions whenever these compatibility conditions
hold. The existence results are proved by using a weighted orthogonal decomposition
theorem for L2
vector fields in terms of scalar and vector potentials. This representation
theorem generalizes the classical HodgeWeyl decomposition. With this special choice
of the potentials, the mixed divcurl 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.
