Sharp Numerical Approximation of the Hardy Constant
Ignat LI, Zuazua Iriondo E (2025)
Publication Language: English
Publication Status: Submitted
Publication Type: Unpublished / Preprint
Future Publication Type: Journal article
Publication year: 2025
Open Access Link: https://dcn.nat.fau.eu/wp-content/uploads/hardy-ignat-zuazua.pdf
Abstract
We study the P1 finite element approximation of the best constant in the classical Hardy inequality over bounded domains containing the origin in R N , 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/| 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.
Authors with CRIS profile
Enrique Zuazua Iriondo
Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)
Involved external institutions
Romania (RO)How to cite
APA:
Ignat, L.I., & Zuazua Iriondo, E. (2025). Sharp Numerical Approximation of the Hardy Constant. (Unpublished, Submitted).
MLA:
Ignat, Liviu I., and Enrique Zuazua Iriondo. Sharp Numerical Approximation of the Hardy Constant. Unpublished, Submitted. 2025.
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