American Institute of Mathematical Sciences

Advanced
May  2008, 7(3): 533-562. doi: 10.3934/cpaa.2008.7.533

Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators

Roberta Bosi 1, , Jean Dolbeault 2, and  Maria J. Esteban 3,

1. 

Institut für Analysis und Scientific Computing, Wiedner Hauptstr. 8, A 1040 Wien, Österreich, Austria
2. 

Ceremade (UMR CNRS no. 7534), Université Paris Dauphine, Place de Lattre de Tassigny, 75775 Paris Cédex 16
3. 

CEREMADE - UMR C.N.R.S. 7534, Université Paris IX-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France

Received  November 2006 Revised  August 2007 Published  February 2008

By expanding squares, we prove several Hardy inequalities with two critical singularities and constants which explicitly depend upon the distance between the two singularities. These inequalities involve the $L^2$ norm. Such results are generalized to an arbitrary number of singularities and compared with standard results given by the IMS method. The generalized version of Hardy inequalities with several singularities is equivalent to some spectral information on a Schrödinger operator involving a potential with several inverse square singularities. We also give a generalized Hardy inequality for Dirac operators in the case of a potential having several singularities of Coulomb type, which are critical for Dirac operators.
Keywords: Hardy inequalities, optimal inequalities, Schrödinger operator, Dirac-Coulomb Hamiltonian., singular potentials, weighted norms.
Mathematics Subject Classification: Primary: 26D10, 35J10, 46E35; Secondary: 35P15, 35Q40, 46N50, 47N50, 81V45, 81V5.
Citation: Roberta Bosi, Jean Dolbeault, Maria J. Esteban. Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators. Communications on Pure and Applied Analysis, 2008, 7 (3) : 533-562. doi: 10.3934/cpaa.2008.7.533
[1]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure and Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274
[2]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1945-1966. doi: 10.3934/dcdss.2020469
[3]

Manli Song, Jinggang Tan. Hardy inequalities for the fractional powers of the Grushin operator. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4699-4726. doi: 10.3934/cpaa.2020192
[4]

Craig Cowan. Optimal Hardy inequalities for general elliptic operators with improvements. Communications on Pure and Applied Analysis, 2010, 9 (1) : 109-140. doi: 10.3934/cpaa.2010.9.109
[5]

Wenxiong Chen, Chao Jin, Congming Li, Jisun Lim. Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations. Conference Publications, 2005, 2005 (Special) : 164-172. doi: 10.3934/proc.2005.2005.164
[6]

Jerome A. Goldstein, Ismail Kombe, Abdullah Yener. A unified approach to weighted Hardy type inequalities on Carnot groups. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2009-2021. doi: 10.3934/dcds.2017085
[7]

Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363
[8]

Barbara Brandolini, Francesco Chiacchio, Cristina Trombetti. Hardy type inequalities and Gaussian measure. Communications on Pure and Applied Analysis, 2007, 6 (2) : 411-428. doi: 10.3934/cpaa.2007.6.411
[9]

Juan Luis Vázquez, Nikolaos B. Zographopoulos. Hardy type inequalities and hidden energies. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5457-5491. doi: 10.3934/dcds.2013.33.5457
[10]

Lorenzo Brasco, Eleonora Cinti. On fractional Hardy inequalities in convex sets. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4019-4040. doi: 10.3934/dcds.2018175
[11]

Biagio Cassano, Lucrezia Cossetti, Luca Fanelli. Improved Hardy-Rellich inequalities. Communications on Pure and Applied Analysis, 2022, 21 (3) : 867-889. doi: 10.3934/cpaa.2022002
[12]

Antonio Cañada, Salvador Villegas. Optimal Lyapunov inequalities for disfocality and Neumann boundary conditions using $L^p$ norms. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 877-888. doi: 10.3934/dcds.2008.20.877
[13]

Ismail Kombe, Abdullah Yener. A general approach to weighted $L^{p}$ Rellich type inequalities related to Greiner operator. Communications on Pure and Applied Analysis, 2019, 18 (2) : 869-886. doi: 10.3934/cpaa.2019042
[14]

Judith Vancostenoble. Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 761-790. doi: 10.3934/dcdss.2011.4.761
[15]

Holger Teismann. The Schrödinger equation with singular time-dependent potentials. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 705-722. doi: 10.3934/dcds.2000.6.705
[16]

Sami Aouaoui, Rahma Jlel. Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 781-813. doi: 10.3934/dcds.2021137
[17]

Angelo Alvino, Roberta Volpicelli, Bruno Volzone. A remark on Hardy type inequalities with remainder terms. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 801-807. doi: 10.3934/dcdss.2011.4.801
[18]

Pradeep Boggarapu, Luz Roncal, Sundaram Thangavelu. On extension problem, trace Hardy and Hardy's inequalities for some fractional Laplacians. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2575-2605. doi: 10.3934/cpaa.2019116
[19]

Antonio Avantaggiati, Paola Loreti, Cristina Pocci. Mixed norms, functional Inequalities, and Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 1855-1867. doi: 10.3934/dcdsb.2014.19.1855
[20]

T. V. Anoop, Nirjan Biswas, Ujjal Das. Admissible function spaces for weighted Sobolev inequalities. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3259-3297. doi: 10.3934/cpaa.2021105

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (79)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy