Existence of solutions to the (rot, div)-system in $L_p$-weighted spaces
Volume 37 / 2010
Applicationes Mathematicae 37 (2010), 127-142
MSC: Primary 35J55.
DOI: 10.4064/am37-2-1
Abstract
The existence of solutions to the elliptic problem $\mathop{\rm rot} v=w$, $\mathop{\rm div} v=0$ in a bounded domain ${\mit\Omega}\subset\Bbb R^3$, $v\cdot\bar n|_S=0$, $S=\partial{\mit\Omega}$ in weighted $L_p$-Sobolev spaces is proved. It is assumed that an axis $L$ crosses $\mit\Omega$ and the weight is a negative power function of the distance to the axis. The main part of the proof is devoted to examining solutions of the problem in a neighbourhood of $L$. The existence in $\mit\Omega$ follows from the technique of regularization.
