Localisation for the torsion function and the strong Hardy inequality

TL;DR

This paper establishes bounds and localization properties of the torsion function using Hardy inequalities, providing detailed analysis and examples to deepen understanding of its behavior in relation to boundary proximity.

Contribution

It introduces new bounds and localization results for the torsion function based on Hardy inequalities, advancing theoretical understanding in this area.

Findings

Derived two-sided bounds for the torsion function efficiency.

Established localization properties under Hardy inequality assumptions.

Analyzed a detailed example illustrating the theoretical results.

Abstract

Two-sided bounds for the efficiency of the torsion function are obtained in terms of the square of the distance to the boundary function under the hypothesis that the Dirichlet Laplacian satisfies a strong Hardy inequality. Localisation properties of the torsion function are obtained under that hypothesis. An example is analysed in detail.