Existence of solutions to the $({\rm rot},{\rm div})$-system in $L_2$-weighted spaces

Volume 36 / 2009

Wojciech M. Zaj/aczkowski Applicationes Mathematicae 36 (2009), 83-106 DOI: 10.4064/am36-1-7

Abstract

The existence of solutions to the elliptic problem $\textrm{ rot } v=w$, $\textrm{div } v=0$ in ${\mit\Omega}\subset\Bbb R^3$, $v\cdot\overline n|_S=0$, $S=\partial\mit\Omega$, in weighted Hilbert spaces is proved. It is assumed that $\mit\Omega$ contains an axis $L$ and the weight is a negative power of the distance to the axis. The main part of the proof is devoted to examining solutions in a neighbourhood of $L$. Their existence in $\mit\Omega$ follows by regularization.

Authors

  • Wojciech M. Zaj/aczkowskiInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-956 Warszawa, Poland
    and
    Institute of Mathematics and Cryptology
    Cybernetics Faculty
    Military University of Technology
    Kaliskiego 2
    00-908 Warszawa, Poland
    e-mail

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