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Exact Poincaré constants in two‐dimensional annuli
Author(s) -
Rummler Bernd,
Růžička Michael,
Thäter Gudrun
Publication year - 2017
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201500299
Subject(s) - laplace operator , eigenvalues and eigenvectors , dirichlet eigenvalue , mathematical analysis , hessian matrix , dirichlet boundary condition , scalar (mathematics) , mathematics , solenoidal vector field , poincaré conjecture , constant (computer programming) , limit (mathematics) , boundary (topology) , vector field , mathematical physics , physics , geometry , quantum mechanics , dirichlet's principle , computer science , programming language
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 Ω A is defined via two concentrical circles with radii A / 2 and A / 2 + 1 . Additionally, corresponding problems on domains Ω σ * , the 2d‐annuli from [7][D. S. Lee, 2002], are investigated ‐ for comparison but also to provide limits for A → 0 . In particular, the Green's function of the Laplacian on Ω σ * with vanishing Dirichlet traces on ∂ Ω σ *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 → ∞ .