Regularity for elliptic systems of differential forms and applications

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 $$\delta \left(Ad\omega \right)+B^{T}d\delta \left(B\omega \right)=\lambda B\omega +f\text{ in }\mathrm{\Omega },$$ with either $\nu \wedge \omega$ and $\nu\wedge \delta \left( B\omega \right)$or$\nu\lrcorner B\omega$and$\nu\lrcorner \left( A d\omega \right)$prescribedon$\partial\Omega.$ 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.