Solvability and regularity for an elliptic system prescribing the curl,   divergence, and partial trace of a vector field on Sobolev-class domains
  We provide a self-contained proof of the solvability and regularity of a Hodge-type elliptic system, wherein the divergence and curl of a vector field are prescribed in an open, bounded, Sobolev-class domain, and either the normal component or the tangential components of the vector field are prescribed on the boundary. The proof is based on a regularity theory for vector elliptic equations set on Sobolev-class domains and with Sobolev-class coefficients. 
C.H. Arthur Cheng and Steve Shkoller
math.AP