Description:
<jats:p>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 <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/zamm201500299-math-0001.png" xlink:title="urn:x-wiley:00442267:media:zamm201500299:zamm201500299-math-0001" /> is defined via two concentrical circles with radii <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/zamm201500299-math-0002.png" xlink:title="urn:x-wiley:00442267:media:zamm201500299:zamm201500299-math-0002" /> and <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/zamm201500299-math-0003.png" xlink:title="urn:x-wiley:00442267:media:zamm201500299:zamm201500299-math-0003" />. Additionally, corresponding problems on domains <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/zamm201500299-math-0004.png" xlink:title="urn:x-wiley:00442267:media:zamm201500299:zamm201500299-math-0004" />, the 2d‐annuli from <jats:ext-link xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#zamm201500299-bib-0007" />, are investigated ‐ for comparison but also to provide limits for <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/zamm201500299-math-0005.png" xlink:title="urn:x-wiley:00442267:media:zamm201500299:zamm201500299-math-0005" />. In particular, the Green's function of the Laplacian on <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/zamm201500299-math-0006.png" xlink:title="urn:x-wiley:00442267:media:zamm201500299:zamm201500299-math-0006" /> with vanishing Dirichlet traces on <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/zamm201500299-math-0007.png" xlink:title="urn:x-wiley:00442267:media:zamm201500299:zamm201500299-math-0007" /> is used to show that for <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/zamm201500299-math-0008.png" xlink:title="urn:x-wiley:00442267:media:zamm201500299:zamm201500299-math-0008" /> 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 <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/zamm201500299-math-0009.png" xlink:title="urn:x-wiley:00442267:media:zamm201500299:zamm201500299-math-0009" />.</jats:p>