-WELL-POSEDNESS OF 3D div-curl BOUNDARY VALUE
GILES AUCHMUTY AND JAMES C. ALEXANDER
Abstract. Criteria for the existence and uniqueness of weak solutions of div-curl bound-
ary-value 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, finite-energy (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 boundary-value problem depends on the
differential topology of the region. The problem is well-posed with a unique solution in
) 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