Computation of sharp estimates of the Poincaré constant on planar domains with piecewise self-similar boundary
Lehel Banjai
Heriot-Watt University, Edinburgh, UKLyonell Boulton
Heriot-Watt University, Edinburgh, UK
Abstract
We establish a strategy for finding sharp upper and lower numerical bounds of the Poincaré constant on a class of planar domains with piecewise self-similar boundary. The approach consists of four main components: W1) tight inner-outer shape interpolation, W2) conformal mapping of the approximate polygonal regions, W3) grad-div system formulation of the spectral problem and W4) computation of the eigenvalue bounds. After describing the method, justifying its validity and determining general convergence estimates, we show concrete evidence of its effectiveness by computing lower and upper bound estimates for the constant on the Koch snowflake.
Cite this article
Lehel Banjai, Lyonell Boulton, Computation of sharp estimates of the Poincaré constant on planar domains with piecewise self-similar boundary. J. Fractal Geom. 8 (2021), no. 2, pp. 153–188
DOI 10.4171/JFG/101