Solvability and regularity for an elliptic system prescribing the curl, divergence, and partial trace of a vector field on Sobolev-class domains

TL;DR

This paper proves the solvability and regularity of a Hodge-type elliptic system for vector fields with prescribed divergence, curl, and boundary conditions on Sobolev-class domains, using a new regularity theory.

Contribution

It provides a self-contained proof of elliptic system solvability and regularity on Sobolev-class domains with Sobolev coefficients, extending previous results.

Findings

Establishes solvability of the elliptic system under Sobolev regularity assumptions

Develops a regularity theory for vector elliptic equations on Sobolev-class domains

Demonstrates boundary condition prescriptions for vector fields in Sobolev spaces

Abstract

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.