Existence of solutions to the $({\rm rot},{\rm div})$-system in $L_2$-weighted spaces
Tom 36 / 2009
Applicationes Mathematicae 36 (2009), 83-106
DOI: 10.4064/am36-1-7
Streszczenie
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.
