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 $$\begin{aligned} \delta \left( A d\omega \right) + B^{T}d\delta \left( B\omega \right) = \lambda B\omega + f \text { in } \varOmega , \end{aligned}$$ 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 $$\partial \varOmega .$$ 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.