In a three-dimensional bounded possibly multiply connected domain, we prove the existence, uniqueness and regularity of some vector potentials, associated with a divergence-free function and satisfying mixed boundary conditions. For such a construction, the fundamental tool is the characterization of the kernel which is related to the topology of the domain. We also give several estimates of vector fields via the operators div and curl when mixing tangential and normal components on the boundary. Furthermore, we establish some Inf-Sup conditions that are crucial in the L p-theory proofs. Finally, we apply the obtained results to solve the Stokes problem with a pressure condition on some part of the boundary and Navier-type boundary condition on the remaining part, where weak and strong solutions are considered.