Regularity for elliptic systems of differential forms and applications

TL;DR

This paper establishes existence and regularity results for solutions to a class of elliptic systems involving differential forms, with applications to Maxwell, Stokes, and div-curl problems, using a Campanato method approach.

Contribution

It introduces a boundary regularity framework for elliptic systems of differential forms that avoids classical potential theory and complex boundary condition verifications.

Findings

Proves regularity estimates in $L^p$ and H"older spaces for weak solutions.

Applies results to Maxwell, Stokes, and div-curl systems.

Uses a Campanato method avoiding potential theory.

Abstract

We prove existence and up to the boundary regularity estimates in $L^{p}$ and H\"{o}lder spaces for weak solutions of the linear system $$ \delta \left( A d\omega \right) + B^{T}d\delta \left( B\omega \right) = \lambda B\omega + f \text{ in } \Omega, $$ with either $ \nu\wedge \omega$ and $\nu\wedge \delta \left( B\omega \right)$ or $\nu\lrcorner B\omega$ and $\nu\lrcorner \left( A d\omega \right)$ prescribed on $\partial\Omega.$ The proofs are in the spirit of `Campanato method' and thus avoid potential theory and do not require a verification of Agmon-Douglis-Nirenberg or Lopatinski\u{i}-Shapiro type conditions. Applications to a number of related problems, such as general versions of the time-harmonic Maxwell system, stationary Stokes problem and the `div-curl' systems, are included.