| Abstract: |
| The main purpose of this paper is to prove the existence, uniqueness and regularity of some vector potentials, associated with a divergence-free function and satisfying mixed boundary conditions in a three-dimensional bounded possibly multiply connected domain. We establish some estimates of vector fields via div and {\bf curl} when tangential and normal boundary conditions vanish \emph{ie.} ${\boldsymbol u}\times {\boldsymbol n} = {\bf 0}$ on some part of the boundary and ${\boldsymbol u}\cdot {\boldsymbol n} = 0$ on the other part. Furthermore, we characterize the kernels and we establish some Inf-Sup conditions which turns out to be the key point of the proofs in $L^{p}$-theory. Finally, we apply the obtained results to develop some regularity properties for the Stokes problem with Navier-type condition on a part of the boundary and a pressure condition on the remaining part.
\KEYWORDS{Vector potentials, mixed boundary conditions, Stokes equations, Navier-type boundary condition.} |
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