Exact Poincare constants in two-dimensional annuli

We provide precise estimates of the Poincare 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 A is defined via two concentrical circles with radii A/2 and A/2+1. Additionally, corresponding problems on domains , the 2d-annuli from , are investigated - for comparison but also to provide limits for 0. In particular, the Green's function of the Laplacian on with vanishing Dirichlet traces on is used to show that for sigma 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.