An explicit right inverse of the divergence operator which is continuous in weighted norms

Ricardo G. Durán, and Maria Amelia Muschietti. "An explicit right inverse of the divergence operator which is continuous in weighted norms." Studia Mathematica 148.3 (2001): 207-219. <http://eudml.org/doc/285268>.

@article{RicardoG2001,
abstract = {The existence of a continuous right inverse of the divergence operator in $W₀^\{1,p\}(Ω)ⁿ$, 1 < p < ∞, is a well known result which is basic in the analysis of the Stokes equations. The object of this paper is to show that the continuity also holds for some weighted norms. Our results are valid for Ω ⊂ ℝⁿ a bounded domain which is star-shaped with respect to a ball B ⊂ Ω. The continuity results are obtained by using an explicit solution of the divergence equation and the classical theory of singular integrals of Calderón and Zygmund together with general results on weighted estimates proven by Stein. The weights considered here are of interest in the analysis of finite element methods. In particular, our result allows us to extend to the three-dimensional case the general results on uniform convergence of finite element approximations of the Stokes equations.},
author = {Ricardo G. Durán, Maria Amelia Muschietti},
journal = {Studia Mathematica},
keywords = {divergence operator; singular integrals; weighted estimates; Stokes equations; finite elements},
language = {eng},
number = {3},
pages = {207-219},
title = {An explicit right inverse of the divergence operator which is continuous in weighted norms},
url = {http://eudml.org/doc/285268},
volume = {148},
year = {2001},
}

TY - JOUR
AU - Ricardo G. Durán
AU - Maria Amelia Muschietti
TI - An explicit right inverse of the divergence operator which is continuous in weighted norms
JO - Studia Mathematica
PY - 2001
VL - 148
IS - 3
SP - 207
EP - 219
AB - The existence of a continuous right inverse of the divergence operator in $W₀^{1,p}(Ω)ⁿ$, 1 < p < ∞, is a well known result which is basic in the analysis of the Stokes equations. The object of this paper is to show that the continuity also holds for some weighted norms. Our results are valid for Ω ⊂ ℝⁿ a bounded domain which is star-shaped with respect to a ball B ⊂ Ω. The continuity results are obtained by using an explicit solution of the divergence equation and the classical theory of singular integrals of Calderón and Zygmund together with general results on weighted estimates proven by Stein. The weights considered here are of interest in the analysis of finite element methods. In particular, our result allows us to extend to the three-dimensional case the general results on uniform convergence of finite element approximations of the Stokes equations.
LA - eng
KW - divergence operator; singular integrals; weighted estimates; Stokes equations; finite elements
UR - http://eudml.org/doc/285268
ER -