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PreprintCC BY 4.0arXiv

Estimates of the Poincare constants for bounded domains in Euclidian space R^n

  • Bernd Rummler
  • Gudrun Thäter
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Abstract

We study bounded, simply connected, convex, and C2-smooth domains in the space R^n for finite n greater than 1. For such domains we use the diameter to find balls with the property that they are included in this ball, while the diameter of the ball is chosen to be minimal. With a simple transformation we are able to map our domain onto a domain inside the open unit ball. We derive estimates for the Poincare constants of scalar and vector functions on those domains. The main ingredient is the first eigenvalue of the Laplacian or the Stokes operator with vanishing Dirichlet traces, respectively (we use results of [21]). Additionally we apply the relation of the first eigenvalues and one eigenfunction of the (scalar) Laplace and the Stokes operator cf. [21]. Our estimates for the Poincare constants are new. The dimension n of the space has significant effect on the quality of our estimates. This is illustrated with the help of an example.

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Cite

BibTeXRIS
BibTeX
@misc{rummler2026estimates,
  title = {{Estimates of the Poincare constants for bounded domains in Euclidian space R^n}},
  author = {Bernd Rummler and Gudrun Thäter},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08028},
  eprint = {2610.08028},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08028},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Estimates of the Poincare constants for bounded domains in Euclidian space R^n
AU  - Bernd Rummler
AU  - Gudrun Thäter
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08028
UR  - https://arxiv.org/abs/2610.08028
AB  - We study bounded, simply connected, convex, and C2-smooth domains in the space R^n for finite n greater than 1. For such domains we use the diameter to find balls with the property that they are included in this ball, while the diameter of the ball is chosen to be minimal. With a simple transformation we are able to map our domain onto a domain inside the open unit ball. We derive estimates for the Poincare constants of scalar and vector functions on those domains. The main ingredient is the first eigenvalue of the Laplacian or the Stokes operator with vanishing Dirichlet traces, respectively (we use results of [21]). Additionally we apply the relation of the first eigenvalues and one eigenfunction of the (scalar) Laplace and the Stokes operator cf. [21]. Our estimates for the Poincare constants are new. The dimension n of the space has significant effect on the quality of our estimates. This is illustrated with the help of an example.
ER  -