Computation of sharp estimates of the Poincaré constant on planar domains with piecewise self-similar boundary

  • Lehel Banjai

    Heriot-Watt University, Edinburgh, UK
  • Lyonell Boulton

    Heriot-Watt University, Edinburgh, UK
Computation of sharp estimates of the Poincaré constant on planar domains with piecewise self-similar boundary cover
Download PDF

This article is published open access under our Subscribe to Open model.

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