Exact Poincaré constants in two-dimensional annuli
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- 作者：Bernd Rummler ; Michael Růžička and Gudrun Thä ; ter
- 刊名：ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik
- 出版年：2017
- 出版时间：January 2017
- 年：2017
- 卷：97
- 期：1
- 页码：110-122
- 全文大小：355K
- ISSN：1521-4001
文摘
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 is defined via two concentrical circles with radii and . Additionally, corresponding problems on domains , the 2d-annuli from [zamm201500299-bib-0007" rel="references:#zamm201500299-bib-0007" class="link__reference js-link__reference" title="Link to bibliographic citation">7], are investigated - for comparison but also to provide limits for . In particular, the Green's function of the Laplacian on with vanishing Dirichlet traces on is used to show that for 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 .