Palladium‐catalyzed reactions in general are carried out under an inert atmosphere because the palladium intermediates involved in the catalytic cycles are often known to be sensitive to oxygen. In this paper, we report that various palladium‐catalyzed reductive couplings proceed smoothly under an air atmosphere and in aqueous medium. Under the air atmosphere reaction conditions, palladium‐triphenylphosphine complex was found to be inactive. By using zinc as the reducing reagent, aryl halides were homo‐coupled to give symmetrical biaryls either in aqueous acetone or in water in the presence of a catalytic amount of 18‐crown‐6 at room temperature. Both aryl iodides and aryl bromides reacted efficiently under the current reaction conditions. The reaction of arylhalosilanes with aryl halides under refluxing conditions in air and water catalyzed by palladium generated unsymmetrical biaryls efficiently in the presence of either KOH or NaF. Such air‐stable couplings are also suitable for a Suzuki‐type coupling and a Stille‐type coupling.
We consider the eigenvalue problem of the Stokes operator in a bounded cylindrical domain of ℝ3 with homogeneous Dirichlet boundary conditions on the curved part of the boundary and periodical conditions in the main stream direction. We deduce by separation the corresponding systems of ordinary differential equations and solve them explicitly looking for bounded, solenoidal vector fields fulfilling the boundary conditions. The investigation of possible cases yields either the explicit eigenfunctions and eigenvalues, or equations for the determination of the eigenvalues, and a general representation of the eigenfunctions.
We investigate the eigenvalue problem of the Stokes operator in an open bounded rectangular parallelepiped of the ℝ3 supplemented with homogeneous Dirichlet boundary conditions on two face to face lying lateral surfaces and periodical conditions in the directions orthogonal to the other faces. By separation we deduce a corresponding system of ordinary differential equations. We look for solutions as solenoidal vector fields satifying the boundary conditions. The investigation of possible cases yields either the explicit eigenfunctions and eigenvalues or equations for determining the eigenvalues and a general deduced form of the eigenfunctions. In the appendix we give an example of the first eigenvalues of the Stokes operator for fixed l = 2.69.
We provide precise estimates of the Poincaré constants firstly for scalar functions and secondly for solenoidal (i.e. divergence free) vector fields (in both cases with vanishing Dirichlet traces on the boundary) on 2d‐annuli by the use of the first eigenvalues of the scalar Laplacian and the Stokes operator, respectively. In our non‐dimensional setting each annulus ΩA is defined via two concentrical circles with radii A/2 and A/2+1. Additionally, corresponding problems on domains Ωσ*, the 2d‐annuli from , are investigated ‐ for comparison but also to provide limits for A→0. In particular, the Green's function of the Laplacian on Ωσ* with vanishing Dirichlet traces on ∂Ωσ* is used to show that for σ→0 the first eigenvalue here tends to the first eigenvalue of the corresponding problem on the open unit circle. On the other hand, we take advantage of the so‐called small‐gap limit for A→∞.
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