Abstract

We consider finite element approximations to the optimal constant for the Hardy inequality with exponent p=2p=2 in bounded domains of dimension n=1n=1 or n≥3n\geq 3. For finite element spaces of piecewise linear and continuous functions on a mesh of size hh, we prove that the approximate Hardy constant converges to the optimal Hardy constant at a rate proportional to 1/∣log⁡h∣21/|\log h|^2. This result holds in dimension n=1n=1, in any dimension n≥3n\geq 3 if the domain is the unit ball and the finite element discretization exploits the rotational symmetry of the problem, and in dimension n=3n=3 for general finite element discretizations of the unit ball. In the first two cases, our estimates show excellent quantitative agreement with values of the discrete Hardy constant obtained computationally.

Author information

Contact details are reproduced from the original publication and may be historical.

Francesco Della Pietra

Dip. di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli "Federico II",
Napoli, Italy

f.dellapietra@unina.it

Giovanni Fantuzzi

Dept. of Mathematics, Friedrich-Alexander-Universität, Erlangen-Nürnberg, Germany

giovanni.fantuzzi@fau.de

Liviu I. Ignat

(1) Institute of Mathematics "Simion Stoilow", Romanian Academy, Bucharest, Romania
(2) Research Institute of the University of Bucharest ICUB, Bucharest, Romania

liviu.ignat@gmail.com

Alba Lia Masiello

Dip. di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli "Federico II",
Napoli, Italy

albalia.masiello@unina.it

Gloria Paoli

Dip. di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli "Federico II",
Napoli, Italy

gloria.paoli@unina.it

Enrique Zuazua

(1) Dept. of Mathematics, Friedrich-Alexander-Universität, Erlangen-Nürnberg, Germany
(2) Chair of Computational Mathematics, Fundación Deusto, Bilbao, Spain
(3) Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain

enrique.zuazua@fau.de

Keywords

  • Hardy inequality
  • Hardy constant
  • finite element method

Mathematics Subject Classification

46E35, 65N30

References

  1. A. Adimurthi, K. Sandeep: Existence and non-existence of the first eigenvalue of the perturbed Hardy-Sobolev operator, Proc. Roy. Soc. Edinburgh Sect. A 132(5) (2002) 1021–1043.https://doi.org/10.1017/S0308210500001992Google Scholar
  2. P. F. Antonietti, A. Pratelli: Finite element approximation of the Sobolev constant, Numer. Math. 117/1 (2011) 37–64.https://doi.org/10.1007/s00211-010-0347-yGoogle Scholar
  3. H. Brezis, M. Marcus: Hardy’s inequalities revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25/1-2 (1998) 217–237.Google Scholar
  4. H. Brezis, J. L. Vázquez: Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10/2 (1997) 443–469.https://doi.org/10.5209/rev_REMA.1997.v10.n2.17459Google Scholar
  5. A. Buffa, C. Ortner: Compact embeddings of broken Sobolev spaces and applications, IMA J. Numer. Anal. 29/4 (2009) 827–855.https://doi.org/10.1093/imanum/drn038Google Scholar
  6. G. Buttazzo, M. Giaquinta, S. Hildebrandt: One-Dimensional Variational Problems, Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford (1998).https://doi.org/10.1093/oso/9780198504658.001.0001Google Scholar
  7. C. Carstensen, J. Gedicke: Guaranteed lower bounds for eigenvalues, Math. Comp. 83/290 (2014) 2605–2629.https://doi.org/10.1090/S0025-5718-2014-02833-0Google Scholar
  8. C. Cazacu: Schrödinger operators with boundary singularities: Hardy inequality, Pohozaev identity and controllability results, J. Funct. Anal. 263/12 (2012) 3741–3783.https://doi.org/10.1016/j.jfa.2012.09.006Google Scholar
  9. A. Chernyavsky, J. J. Bramburger, G. Fantuzzi, D. Goluskin: Convex relaxations of integral variational problems: pointwise dual relaxation and sum-of-squares optimization, SIAM J. Optim. 33/2 (2023) 481–512.https://doi.org/10.1137/21M1455127Google Scholar
  10. M. del Pino, J. Dolbeault, S. Filippas, A. Tertikas: A logarithmic Hardy inequality, J. Funct. Anal. 259/8 (2010) 2045–2072.https://doi.org/10.1016/j.jfa.2010.06.005Google Scholar
  11. F. Della Pietra, G. di Blasio, N. Gavitone: Anisotropic Hardy inequalities, Proc. Roy. Soc. Edinburgh Sect. A 148/3 (2018) 483–498.https://doi.org/10.1017/S0308210517000336Google Scholar
  12. G. Fantuzzi, I. Tobasco: Sharpness and non-sharpness of occupation measure bounds for integral variational problems, arXiv: 2207.13570 (2022),Google Scholar
  13. C. L. Fefferman: The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9/2 (1983) 129–206.https://doi.org/10.1090/S0273-0979-1983-15154-6Google Scholar
  14. W. M. Frank, D. J. Land, R. M. Spector: Singular potentials, Rev. Modern Phys. 43/1 (1971) 36–98.https://doi.org/10.1103/RevModPhys.43.36Google Scholar
  15. N. Fusco, F. Maggi, A. Pratelli: The sharp quantitative Sobolev inequality for functions of bounded variation, J. Funct. Anal. 244/1 (2007) 315–341.https://doi.org/10.1016/j.jfa.2006.10.015Google Scholar
  16. J. P. García Azorero, I. Peral Alonso: Hardy inequalities and some critical elliptic and parabolic problems, J. Diff. Equations 144/2 (1998) 441–476.https://doi.org/10.1006/jdeq.1997.3375Google Scholar
  17. F. Gazzola, H.-C. Grunau, E. Mitidieri: Hardy inequalities with optimal constants and remainder terms, Trans. Amer. Math. Soc. 356/6 (2004) 2149–2168.https://doi.org/10.1090/S0002-9947-03-03395-6Google Scholar
  18. S. Guo, X. Lu, Z. Zhang: Finite element method for an eigenvalue optimization problem of the Schrödinger operator, AIMS Math. 7/4 (2022) 5049–5071.https://doi.org/10.3934/math.2022281Google Scholar
  19. G. H. Hardy: Note on a theorem of Hilbert, Math. Z. 6/3-4 (1920) 314–317.https://doi.org/10.1007/BF01199965Google Scholar
  20. G. H. Hardy, J. E. Littlewood, G. Pólya: Inequalities, Cambridge University Press, Cambridge (1988).Google Scholar
  21. D. Henrion, M. Korda, M. Kružík, R. Rios-Zertuche: Occupation measure relaxations in variational problems: the role of convexity, arXiv: 2303.02434 (2023).Google Scholar
  22. M. Korda, D. Henrion, J. B. Lasserre: Moments and convex optimization for analysis and control of nonlinear PDEs, in: Numerical Control, Part A, Handbook of Numerical Analysis Vol. 23, North-Holland, Amsterdam (2022) 339–366.https://doi.org/10.1016/bs.hna.2021.12.010Google Scholar
  23. D. Krejčiřík, E. Zuazua: The Hardy inequality and the heat equation in twisted tubes, J. Math. Pures Appl. (9) 94/3 (2010) 277–303.https://doi.org/10.1016/j.matpur.2010.02.006Google Scholar
  24. A. Kufner, L. Maligranda, L.-E. Persson: The prehistory of the Hardy inequality, Amer. Math. Monthly 113/8 (2006) 715–732.https://doi.org/10.1080/00029890.2006.11920356Google Scholar
  25. A. Kufner, L. Maligranda, L.-E. Persson: The Hardy Inequality, Vydavatelský Servis, Plzeň (2007).Google Scholar
  26. A. Kufner, L.-E. Persson, N. Samko: Weighted Inequalities of Hardy Type, 2nd ed., World Scientific, Hackensack (2017).https://doi.org/10.1142/10052Google Scholar
  27. I. Peral Alonso, F. Soria de Diego: Elliptic and Parabolic Equations Involving the Hardy-Leray Potential, De Gruyter Series in Nonlinear Analysis and Applications 38, Walter de Gruyter, Berlin (2021).https://doi.org/10.1515/9783110606270Google Scholar
  28. A. Quarteroni, A. Valli: Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics 23, Springer, Berlin (1994).https://doi.org/10.1007/978-3-540-85268-1Google Scholar
  29. P.-A. Raviart, J.-M. Thomas: Introduction à l’Analyse Numérique des Équations aux Dérivées Partielles, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris (1983).Google Scholar
  30. Y. Sui, D. Zhang, J. Cao, J. Zhang: An efficient finite element method and error analysis for eigenvalue problem of Schrödinger equation with an inverse square potential on spherical domain, Adv. Difference Equations (2020), art. no. 582, 14 p.https://doi.org/10.1186/s13662-020-03034-9Google Scholar
  31. J. L. Vázquez, E. Zuazua: The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal. 173/1 (2000) 103–153.https://doi.org/10.1006/jfan.1999.3556Google Scholar
  32. Z.-Q. Wang, M. Willem: Caffarelli-Kohn-Nirenberg inequalities with remainder terms, J. Funct. Anal. 203/2 (2003) 550–568.https://doi.org/10.1016/S0022-1236(03)00017-XGoogle Scholar
  33. H. F. Weinberger. Upper and lower bounds for eigenvalues by finite difference methods, Comm. Pure Appl. Math. 9 (1956) 613–623.https://doi.org/10.1002/cpa.3160090329Google Scholar

Suggested citation

F. Della Pietra, G. Fantuzzi, L. I. Ignat, A. L. Masiello, G. Paoli, E. Zuazua. “Finite Element Approximation of the Hardy Constant.” Journal of Convex Analysis 31 (2024), No. 2, 497–523. https://doi.org/10.68381/jca31027

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2024. Rights now held by Banach Press.