American Institute of Mathematical Sciences

Advanced
May  2014, 34(5): 1951-1959. doi: 10.3934/dcds.2014.34.1951

An extended discrete Hardy-Littlewood-Sobolev inequality

Ze Cheng 1, and  Congming Li 1,

1. 

Department of Applied Mathematics, University of Colorado at Boulder, Colorado, United States, United States

Received  May 2013 Revised  July 2013 Published  October 2013

Hardy-Littlewood-Sobolev (HLS) Inequality fails in the ``critical'' case: $μ=n$. However, for discrete HLS, we can derive a finite form of HLS inequality with logarithm correction for a critical case: $μ=n$ and $p=q$, by limiting the inequality on a finite domain. The best constant in the inequality and its corresponding solution, the optimizer, are studied. First, we obtain a sharp estimate for the best constant. Then for the optimizer, we prove the uniqueness and a symmetry property. This is achieved by proving that the corresponding Euler-Lagrange equation has a unique nontrivial nonnegative critical point. Also, by using a discrete version of maximum principle, we prove certain monotonicity of this optimizer.
Keywords: HLS inequality, Euler-Lagrange equation., maximum principle.
Mathematics Subject Classification: Primary: 35A2.
Citation: Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951
References:
[1]

W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev Inequalities and Systems of Integral Equations,, Discrete and Continuous Dynamical Systems, (2005), 164. Google Scholar

[2]

W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Discrete Contin. Dyn. Syst., 24 (2009), 1167. doi: 10.3934/dcds.2009.24.1167. Google Scholar

[3]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8. Google Scholar

[4]

W. Chen and C. Li, Indefinite elliptic problems in a domain,, Discrete Contin. Dynam. Systems, 3 (1997), 333. doi: 10.3934/dcds.1997.3.333. Google Scholar

[5]

W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, Discrete Contin. Dyn. Syst., 30 (2011), 1083. doi: 10.3934/dcds.2011.30.1083. Google Scholar

[6]

W. Chen and C. Li, Regularity of solutions for a system of integral equations,, Comm Pure Appl Anal, 4 (2005), 1. Google Scholar

[7]

W. Chen and C. Li, The best constant in some weighted Hardy-Littlewood-Sobolev inequality,, Proc. AMS, 136 (2008), 955. doi: 10.1090/S0002-9939-07-09232-5. Google Scholar

[8]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Commun. in Partial Differential Equations, 30 (2005), 59. doi: 10.1081/PDE-200044445. Google Scholar

[9]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Disc. & Cont. Dynamics Sys., 12 (2005), 347. Google Scholar

[10]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure and Appl. Math., 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar

[11]

L. E. Fraenkel, An introduction to maximum principles and symmetry in elliptic problems,, Cambridge Tracts in Mathematics, (2000). doi: 10.1017/CBO9780511569203. Google Scholar

[12]

B. Gidas, W. Ni and L. Nirenberg, Symmetry of Positive Solutions of Nonlinear Elliptic Equations in $R^n$,, Mathematical Analysis and Applications, (1981), 369. Google Scholar

[13]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality,, Math Res Lett, 14 (2007), 373. doi: 10.4310/MRL.2007.v14.n3.a2. Google Scholar

[14]

C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. Amer. Math. Soc., 134 (2006), 1661. doi: 10.1090/S0002-9939-05-08411-X. Google Scholar

[15]

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities,, Cambridge at the University Press, (1952). Google Scholar

[16]

J. Kigami, Analysis on Fractals,, Cambridge Tracts in Mathematics, (2001). doi: 10.1017/CBO9780511470943. Google Scholar

[17]

Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations,, Calc. Var. Partial Differential Equations, 45 (2012), 43. doi: 10.1007/s00526-011-0450-7. Google Scholar

[18]

Y. Lei, C. Li and C. Ma, Decay estimation for positive solutions of a $\gamma$-Laplace equation,, Discrete Contin. Dyn. Syst., 30 (2011), 547. doi: 10.3934/dcds.2011.30.547. Google Scholar

[19]

C. Li and J. Lim, The singularity analysis of solutions to some integral equations,, Commun. Pure Appl. Anal, 6 (2007), 453. doi: 10.3934/cpaa.2007.6.453. Google Scholar

[20]

C. Li and J. Villavert, An extension of the Hardy-Littlewood-Pólya inequality,, Acta Mathematica Scientia, 31 (2011), 2285. doi: 10.1016/S0252-9602(11)60400-1. Google Scholar

[21]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math., 118 (1983), 349. doi: 10.2307/2007032. Google Scholar

[22]

O. Perron, Zur Theorie der Matrices,, Mathematische Annalen, 64 (1907), 248. doi: 10.1007/BF01449896. Google Scholar

[23]

E. B. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space,, J. Math. Mech., 7 (1958), 503. doi: 10.1512/iumj.1958.7.57030. Google Scholar

show all references

References:
[1]

W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev Inequalities and Systems of Integral Equations,, Discrete and Continuous Dynamical Systems, (2005), 164. Google Scholar

[2]

W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Discrete Contin. Dyn. Syst., 24 (2009), 1167. doi: 10.3934/dcds.2009.24.1167. Google Scholar

[3]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8. Google Scholar

[4]

W. Chen and C. Li, Indefinite elliptic problems in a domain,, Discrete Contin. Dynam. Systems, 3 (1997), 333. doi: 10.3934/dcds.1997.3.333. Google Scholar

[5]

W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type,, Discrete Contin. Dyn. Syst., 30 (2011), 1083. doi: 10.3934/dcds.2011.30.1083. Google Scholar

[6]

W. Chen and C. Li, Regularity of solutions for a system of integral equations,, Comm Pure Appl Anal, 4 (2005), 1. Google Scholar

[7]

W. Chen and C. Li, The best constant in some weighted Hardy-Littlewood-Sobolev inequality,, Proc. AMS, 136 (2008), 955. doi: 10.1090/S0002-9939-07-09232-5. Google Scholar

[8]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Commun. in Partial Differential Equations, 30 (2005), 59. doi: 10.1081/PDE-200044445. Google Scholar

[9]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Disc. & Cont. Dynamics Sys., 12 (2005), 347. Google Scholar

[10]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure and Appl. Math., 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar

[11]

L. E. Fraenkel, An introduction to maximum principles and symmetry in elliptic problems,, Cambridge Tracts in Mathematics, (2000). doi: 10.1017/CBO9780511569203. Google Scholar

[12]

B. Gidas, W. Ni and L. Nirenberg, Symmetry of Positive Solutions of Nonlinear Elliptic Equations in $R^n$,, Mathematical Analysis and Applications, (1981), 369. Google Scholar

[13]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality,, Math Res Lett, 14 (2007), 373. doi: 10.4310/MRL.2007.v14.n3.a2. Google Scholar

[14]

C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, Proc. Amer. Math. Soc., 134 (2006), 1661. doi: 10.1090/S0002-9939-05-08411-X. Google Scholar

[15]

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities,, Cambridge at the University Press, (1952). Google Scholar

[16]

J. Kigami, Analysis on Fractals,, Cambridge Tracts in Mathematics, (2001). doi: 10.1017/CBO9780511470943. Google Scholar

[17]

Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations,, Calc. Var. Partial Differential Equations, 45 (2012), 43. doi: 10.1007/s00526-011-0450-7. Google Scholar

[18]

Y. Lei, C. Li and C. Ma, Decay estimation for positive solutions of a $\gamma$-Laplace equation,, Discrete Contin. Dyn. Syst., 30 (2011), 547. doi: 10.3934/dcds.2011.30.547. Google Scholar

[19]

C. Li and J. Lim, The singularity analysis of solutions to some integral equations,, Commun. Pure Appl. Anal, 6 (2007), 453. doi: 10.3934/cpaa.2007.6.453. Google Scholar

[20]

C. Li and J. Villavert, An extension of the Hardy-Littlewood-Pólya inequality,, Acta Mathematica Scientia, 31 (2011), 2285. doi: 10.1016/S0252-9602(11)60400-1. Google Scholar

[21]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math., 118 (1983), 349. doi: 10.2307/2007032. Google Scholar

[22]

O. Perron, Zur Theorie der Matrices,, Mathematische Annalen, 64 (1907), 248. doi: 10.1007/BF01449896. Google Scholar

[23]

E. B. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space,, J. Math. Mech., 7 (1958), 503. doi: 10.1512/iumj.1958.7.57030. Google Scholar

[1]

Giovanni Bonfanti, Arrigo Cellina. The validity of the Euler-Lagrange equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 511-517. doi: 10.3934/dcds.2010.28.511
[2]

Stefano Bianchini. On the Euler-Lagrange equation for a variational problem. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 449-480. doi: 10.3934/dcds.2007.17.449
[3]

Menita Carozza, Jan Kristensen, Antonia Passarelli di Napoli. On the validity of the Euler-Lagrange system. Communications on Pure & Applied Analysis, 2015, 14 (1) : 51-62. doi: 10.3934/cpaa.2015.14.51
[4]

Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987
[5]

Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577
[6]

Fabio Paronetto. A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 853-866. doi: 10.3934/dcdss.2017043
[7]

H. O. Fattorini. The maximum principle in infinite dimension. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557
[8]

Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499
[9]

Torsten Lindström. Discrete models and Fisher's maximum principle in ecology. Conference Publications, 2003, 2003 (Special) : 571-579. doi: 10.3934/proc.2003.2003.571
[10]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013
[11]

Chiun-Chuan Chen, Li-Chang Hung, Hsiao-Feng Liu. N-barrier maximum principle for degenerate elliptic systems and its application. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 791-821. doi: 10.3934/dcds.2018034
[12]

Yunkyong Hyon, Do Young Kwak, Chun Liu. Energetic variational approach in complex fluids: Maximum dissipation principle. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1291-1304. doi: 10.3934/dcds.2010.26.1291
[13]

Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control & Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195
[14]

Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial & Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067
[15]

Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581
[16]

H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77
[17]

Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 335-366. doi: 10.3934/cpaa.2007.6.335
[18]

Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395
[19]

Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control & Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022
[20]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences