Abstract
We prove existence and up to the boundary regularity estimates in \(L^{p}\) and Hölder spaces for weak solutions of the linear system
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.
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Acknowledgements
The author thanks Bernard Dacorogna, Jan Kristensen and Hoài-Minh Nguyên for helpful comments and discussions. This work was conceived as a part of author’s doctoral thesis in EPFL, whose support and facilities are also gratefully acknowledged. The author thanks the anonymous referee for the comments and suggestions.
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Communicated by L. Ambrosio.
