Reflection principles and kernels in $\mathbb R^n_+$ for the biharmonic and Stokes operators. Solutions in a large class of weighted Sobolev spaces

Abstract

In this paper, we study the Stokes system in the half-space $\mathbb{R}^n_+$, with $n {\geqslant} 2$. We consider data and give solutions which live in weighted Sobolev spaces, for a whole scale of weights. We start to study the kernels of the biharmonic and Stokes operators. After the central case of the generalized solutions, we are interested in strong solutions and symmetrically in very weak solutions by means of a duality argument.