 
Summary: L2
WELLPOSEDNESS OF 3D divcurl BOUNDARY VALUE
PROBLEMS
GILES AUCHMUTY AND JAMES C. ALEXANDER
Abstract. Criteria for the existence and uniqueness of weak solutions of divcurl bound
aryvalue problems on bounded regions in space with C2
boundaries are developed. The
boundary conditions are either given normal component of the field or else given tan
gential components of the field.
Under natural integrability assumptions on the data, finiteenergy (L2
) solutions exist
if and only if certain compatibility conditions hold on the data. When compatibility
holds, the dimension of the solution space of the boundaryvalue problem depends on the
differential topology of the region. The problem is wellposed with a unique solution in
L2
(; R3
) 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 L2
vector fields
