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

Abstract

The existence of a continuous right inverse of the divergence operator in W01'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 give a constructive proof of the existence of such an operator and to show that the continuity holds also for some weighted norms. Our results are valid for Ω C IP.π a bounded domain which is star-shaped with respect to a ball B C Ω. The continuity results are obtained by using the classical theory of singular integrals of Calderon and Zygmund and 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 to extend to the three dimensional case the general results on uniform convergence of finite element approximations of the Stokes equations.