Sharp numerical approximation of the Hardy constant

TL;DR

This work studies the finite element approximation of the best Hardy constant $\Lambda_N=(N-2)^2/4$ for bounded domains $\Omega$ containing the origin in $\mathbb{R}^N$, where the constant is not attained in $H^1_0(\Omega)$. By leveraging an improved Hardy inequality with a logarithmic remainder and constructing near-minimizers that concentrate at the origin, the authors prove a sharp dimension-independent convergence rate $\Lambda_h(\Omega)-\Lambda_N\simeq 1/|\log h|^2$ for continuous piecewise linear FE spaces. They also analyze related weighted spectral problems, obtaining corresponding logarithmic or near-logarithmic convergence rates, with explicit behavior in the unit ball. The results illuminate the subtle interplay between non-attainment, singular minimizing sequences, and FE discretization, and they open avenues for higher-order methods and adaptive strategies in singular variational problems.

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.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 3.1
• Lemma 3.2
• Lemma 4.1
• Remark 4.2
• proof
• Lemma 8.1
• proof