The Stokes operator in weighted \(L^q\)-spaces. I: Weighted estimates for the Stokes resolvent problem in a half space
The Stokes resolvent problem \[ \lambda v-\Delta v+\nabla p=f, \quad \text{div\,} v=g \quad \text{in \;} \mathbb{R}^n_+,\qquad v=0 \quad \text{on \;} \partial\mathbb{R}^n_+ \] is considered in the half space \(\mathbb{R}^n_+\). The parameter \(\lambda\) belongs to the sector \[ \Sigma_{\varepsilon}= \{\lambda\in C\setminus\{0\}:\;| \text{arg}\,\lambda| <\pi-\varepsilon\},\quad 0<\varepsilon<\frac{\pi}2. \] The solvability and uniqueness of the solution is established in weighted Muckenhoupt classes. Precise estimates to solutions are obtained. These estimates are the base to study \(L^p\)-regularity of the Stokes operator in a bounded domain in a forthcoming paper of the author.
scientific article; zbMATH DE number 1971865