An $L^q(L^2)$-theory of the generalized Stokes resolvent system in infinite cylinders

Reinhard Farwig ¹   ; Myong-Hwan Ri ²

¹ Department of Mathematics Darmstadt University of Technology 64289 Darmstadt, Germany
² Institute of Mathematics Academy of Sciences Pyongyang, DPR Korea

Studia Mathematica, Tome 178 (2007) no. 3, pp. 197-216

Résumé

Estimates of the generalized Stokes resolvent system, i.e. with prescribed divergence, in an infinite cylinder ${\mit\Omega}={\mit\Sigma}\times\mathbb R$ with ${\mit\Sigma}\subset \mathbb R^{n-1}$, a bounded domain of class $C^{1,1}$, are obtained in the space $L^q(\mathbb R;L^2({\mit\Sigma}))$, $q\in (1,\infty)$. As a preparation, spectral decompositions of vector-valued homogeneous Sobolev spaces are studied. The main theorem is proved using the techniques of Schauder decompositions, operator-valued multiplier functions and $R$-boundedness of operator families.

Détail

DOI : 10.4064/sm178-3-1

Keywords: estimates generalized stokes resolvent system prescribed divergence infinite cylinder mit omega mit sigma times mathbb mit sigma subset mathbb n bounded domain class nbsp obtained space mathbb mit sigma infty preparation spectral decompositions vector valued homogeneous sobolev spaces studied main theorem proved using techniques schauder decompositions operator valued multiplier functions r boundedness operator families