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 u are prescribed in an open, bounded, Sobolev-class domain Ω ⊆ Rⁿ, and either the normal component u· N or the tangential components of the vector field u× N are prescribed on the boundary ∂Ω. For k > n / 2 , we prove that u is in the Sobolev space H^{k + 1}(Ω) if Ω is an H^{k + 1}-domain, and the divergence, curl, and either the normal or tangential trace of u has sufficient regularity. The proof is based on a regularity theory for vector elliptic equations set on Sobolev-class domains and with Sobolev-class coefficients, and with a rather general set of Dirichlet and Neumann boundary conditions. The resulting regularity theory for the vector u is fundamental in the analysis of free-boundary and moving interface problems in fluid dynamics.