Regularity for elliptic systems of differential forms and applications

Abstract

We prove existence and up to the boundary regularity estimates in $L^p$ and Hölder spaces for weak solutions of the linear system $$\delta \left(Ad\omega \right)+B^{T}d\delta \left(B\omega \right)=\lambda B\omega +f\text{ in }\Omega ,$$ with either $\nu \wedge \omega$ and $\nu \wedge \delta \left(B\omega \right)$ or $\nu ┘B\omega$ and $\nu ┘\left(Ad\omega \right)$ 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 Lopatinskiĭ-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.

Cited by in corpus (1)

• Quasiconvexity, null Lagrangians, and Hardy space integrability under constant rank constraints