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

Volume 148 / 2001

Ricardo G. Durán, Maria Amelia Muschietti Studia Mathematica 148 (2001), 207-219 MSC: Primary 35Q30, 42B20; Secondary 65N30. DOI: 10.4064/sm148-3-2


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.


  • Ricardo G. DuránDepartamento de Matemática
    Universidad de San Andrés
    Vito Dumas 284
    1644 Victoria
    Provincia de Buenos Aires, Argentina
  • Maria Amelia MuschiettiDepartamento de Matemática
    Facultad de Ciencias Exactas
    Universidad Nacional de La Plata
    Casilla de Correo 172
    1900 La Plata
    Provincia de Buenos Aires, Argentina

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