Sharp Numerical Approximation of the Hardy Constant

Ignat LI, Zuazua Iriondo E (2025)

Publication Language: English

Publication Status: Published

Publication Type: Journal article, Original article

Future Publication Type: Journal article

Publication year: 2025

Journal

Discrete and Continuous Dynamical Systems American Institute of Mathematical Sciences (AIMS)

Publisher: Discrete and Continuous Dynamical Systems (DCDS)

DOI: 10.3934/dcds.2025165

Open Access Link: https://doi.org/10.48550/arXiv.2506.19422

Abstract

We study the P1 finite element approximation of the best constant in the classical Hardy inequality over bounded domains containing the origin in ℝN, for N≥3.
Despite the fact that this constant is not attained in the associated Sobolev space H1, our main result establishes an explicit, sharp, and dimension-independent rate of convergence proportional to 1/|logh|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.

Authors with CRIS profile

Enrique Zuazua Iriondo Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)

Involved external institutions

Romanian Academy / Academia Română RO Romania (RO)

How to cite

APA:

Ignat, L.I., & Zuazua Iriondo, E. (2025). Sharp Numerical Approximation of the Hardy Constant. Discrete and Continuous Dynamical Systems. https://doi.org/10.3934/dcds.2025165

MLA:

Ignat, Liviu I., and Enrique Zuazua Iriondo. "Sharp Numerical Approximation of the Hardy Constant." Discrete and Continuous Dynamical Systems (2025).

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