Sharp numerical approximation of the Hardy constant

TL;DR

This paper develops a precise numerical method to approximate the Hardy constant in bounded domains, achieving a sharp convergence rate that is independent of the dimension, by combining advanced inequalities and finite element analysis.

Contribution

It introduces a novel, explicit convergence rate for finite element approximation of the Hardy constant, which is sharp and dimension-independent, using a combination of improved inequalities and spectral analysis.

Findings

Convergence rate proportional to 1/|log h|^2 established

Finite element approximation effectively captures the Hardy constant

Results extend to related spectral problems with singular potentials

Abstract

We study the $P_1$ finite element approximation of the best constant in the classical Hardy inequality over bounded domains containing the origin in $\mathbb{R}^N$, for $N \geq 3$. Despite the fact that this constant is not attained in the associated Sobolev space $H^1$, our main result establishes an explicit, sharp, and dimension-independent rate of convergence proportional to $1/|\log h|^2$. The analysis carefully combines an improved Hardy inequality involving a reminder term with logarithmic weights, approximation estimates for Hardy-type singular radial functions constituting minimizing sequences, properties of piecewise linear and continuous finite elements, and weighted Sobolev space techniques. We also consider other closely related spectral problems involving the Laplacian with singular quadratic potentials obtaining sharp convergence rates.