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

Ricardo G. Durán ¹   ; Maria Amelia Muschietti ²

¹ Departamento de Matemática Universidad de San Andrés Vito Dumas 284 1644 Victoria Provincia de Buenos Aires, Argentina
² Departamento de Matemática Facultad de Ciencias Exactas Universidad Nacional de La Plata Casilla de Correo 172 1900 La Plata Provincia de Buenos Aires, Argentina

Studia Mathematica, Tome 148 (2001) no. 3, pp. 207-219

Résumé

The existence of a continuous right inverse of the divergence operator in $W^{1,p}_0({\mit\Omega})^n$, $1 p \infty$, 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 ${\mit\Omega}\subset\mathbb R^n$ a bounded domain which is star-shaped with respect to a ball $B\subset{\mit\Omega}$. 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.

Détail

DOI : 10.4064/sm148-3-2

Keywords: existence continuous right inverse divergence operator mit omega infty known result which basic analysis stokes equations object paper continuity holds weighted norms results valid mit omega subset mathbb bounded domain which star shaped respect ball subset mit omega continuity results obtained using explicit solution divergence equation classical theory singular integrals calder zygmund together general results weighted estimates proven stein weights considered here interest analysis finite element methods particular result allows extend three dimensional general results uniform convergence finite element approximations stokes equations