American Institute of Mathematical Sciences

Advanced
May  2012, 11(3): 1363-1386. doi: 10.3934/cpaa.2012.11.1363

Improving sharp Sobolev type inequalities by optimal remainder gradient norms

Andrea Cianchi 1, and  Adele Ferone 2,

1. 

Dipartimento di Matematica "U.Dini", Università di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy
2. 

Dipartimento di Matematica, Seconda Università di Napoli, Viale Lincoln 5, 81100 Caserta, Italy

Received  December 2010 Revised  March 2011 Published  December 2011

We are concerned with Sobolev type inequalities in $W^{1,p}_0(\Omega )$, $\Omega \subset R^n$, with optimal target norms and sharp constants. Admissible remainder terms depending on the gradient are characterized. As a consequence, the strongest possible remainder norm of the gradient is exhibited. Both the case when $p< n$ and the borderline case when $p = n$ are considered. Related Hardy inequalities with remainders are also derived.
Keywords: Hardy-Sobolev inequality, Lorentz spaces., rearrangement invariant spaces, remainder terms.
Mathematics Subject Classification: Primary: 46E35, 46E3.
Citation: Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363
References:
[1]

Adimurthi, N. Chaudhuri and M. Ramaswamy, An improved Hardy-Sobolev inequality and its applications,, Proc. Amer. Math. Soc., 130 (2002), 489.  doi: 10.1090/S0002-9939-01-06132-9.  Google Scholar

[2]

Adimurthi and M. J. Esteban, An improved Hardy-Sobolev inequality in $W^{1,p}$ and its applications to Schrödinger operators,, NoDEA Nonlinear Differential Equation Appl., 12 (2005), 243.  doi: 10.1007/S00030-005-0009-4.  Google Scholar

[3]

A. Alvino, Sulla disuguaglianza di Sobolev in spazi di Lorentz,, Boll. Un. Mat. Ital., 5 (1977), 148.   Google Scholar

[4]

A. Alvino, P.-L.Lions and G. Trombetti, On optimization problems with prescribed rearrangements,, Nonlinear Anal., 13 (1989), 185.  doi: 10.1016/0362-546X(89)90043-6.  Google Scholar

[5]

A. Alvino, R. Volpicelli and B. Volzone, On Hardy inequalities with a remainder term,, Ric. Mat., 59 (2010), 265.  doi: 10.1007/s11587-010-0086-5.  Google Scholar

[6]

T. Aubin, Problèmes isopérimetriques et espaces de Sobolev,, J. Diff. Geom., 11 (1976), 573.   Google Scholar

[7]

G. Barbatis, S. Filippas and A. Tertikas, Series expansion for $L^p$ Hardy inequalities,, Indiana Univ. Math. J., 52 (2003), 171.  doi: 0.1512/iumj.2003.52.2207.  Google Scholar

[8]

G. Barbatis, S. Filippas and A. Tertikas, A unified approach to improved $L^p$ Hardy inequalities with best constants,, Trans. Amer. Math. Soc., 356 (2004), 2169.  doi: 10.1090/S0002-9947-03-03389-0.  Google Scholar

[9]

R. Benguria, R. Frank and M. Loss, The sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half-space,, Math. Res. Lett., 15 (2008), 613.   Google Scholar

[10]

C. Bennett and R. Sharpley, "Interpolation of Operators,", Academic Press, (1988).   Google Scholar

[11]

H.Brezis and E.Lieb, Sobolev inequalities with remainder terms,, J. Funct. Anal., 62 (1985), 73.  doi: 10.1016/0022-1236(85)90020-5.  Google Scholar

[12]

H. Brezis and M. Marcus, Hardy's inequalities revisited,, Ann. Sc. Norm. Super. Pisa, 25 (1997), 217.   Google Scholar

[13]

H. Brezis, M. Marcus and I. Shafrir, Extremal functions for Hardy's inequality with weight,, J. Funct. Anal., 171 (2000), 177.  doi: 10.1006/jfan.1999.3504.  Google Scholar

[14]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437.  doi: 10.1002/cpa.3160360405.  Google Scholar

[15]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities,, Comm. Part. Diff. Eq., 5 (1980), 773.  doi: 10.1080/03605308008820154.  Google Scholar

[16]

J. E. Brothers and W. P. Ziemer, Minimal rearrangements of Sobolev functions,, J. Reine Angew. Math., 384 (1988), 419.  doi: 10.1515/crll.1988.384.153.  Google Scholar

[17]

A. Cianchi and A. Ferone, Best remainder norms in Sobolev-Hardy inequalities,, Indiana Univ. Math. J., 58 (2009), 1051.  doi: 10.1512/iumj.2009.58.3561.  Google Scholar

[18]

G. Crasta, I. Fragalà and F. Gazzola, Some estimates for the torsional rigidity of composite rods,, Math. Nachr., 280 (2007), 242.  doi: 0.1002/mana.200410478.  Google Scholar

[19]

M. Cwikel and E. Pustylnik, Sobolev type embeddings in the limiting case,, J. Fourier Anal. Appl., 4 (1998), 433.  doi: 10.1007/BF02498218.  Google Scholar

[20]

A. Detalla, T. Horiuchi and H. Ando, Missing terms in Hardy-Sobolev inequalities and its applications,, Far East J. Math. Sci., 14 (2004), 333.   Google Scholar

[21]

J. Dolbeault, M. J. Esteban, M. Loss and L. Vega, An analytical proof of Hardy-like inequalities related to the Dirac operator,, J. Funct. Anal., 216 (2004), 1.  doi: 10.1016/j.jfa.2003.09.010.  Google Scholar

[22]

D. E. Edmunds, R. A. Kerman and L. Pick, Optimal Sobolev imbeddings involving rearrangement invariant quasi-norms,, J. Funct. Anal., 170 (2000), 307.  doi: 10.1006/jfan.1999.3508.  Google Scholar

[23]

S. Filippas, V. G. Maz'ya and A. Tertikas, On a question of Brezis and Marcus,, Calc. Var. Partial Differential Equations, 25 (2006), 491.  doi: 10.1007/s00526-005-0353-6.  Google Scholar

[24]

S. Filippas, V. G. Maz'ya and A. Tertikas, Critical Hardy-Sobolev inequalities,, J. Math. Pures Appl., 87 (2007), 37.  doi: 10.1016/j.matpur.2006.10.007.  Google Scholar

[25]

F. Gazzola, H. C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms,, Trans. Amer. Math. Soc., 356 (2004), 2149.  doi: 10.1090/S0002-9947-03-03395-6.  Google Scholar

[26]

N. Ghoussoub and A. Moradifam, On the best possible remaining term in the Hardy inequality,, Proc. Natl. Acad. Sci. USA, 105 (2008), 13746.  doi: 10.1073/pnas.0803703105.  Google Scholar

[27]

E. Giarrusso and D. Nunziante, Symmetrization in a class of first-order Hamilton-Jacobi equations,, Nonlinear Anal., 8 (1984), 289.  doi: 10.1016/0362-546X(84)90031-2.  Google Scholar

[28]

A. Gogatishvili, B. Opic and L. Pick, Weighted inequalities for Hardy-type operators involving suprema,, Collect. Math., 57 (2006), 227.   Google Scholar

[29]

A. Gogatishvili and L. Pick, Discretization and antidiscretization of rearrangement-invariant norms,, Publ. Mat., 47 (2003), 311.   Google Scholar

[30]

K. Hansson, Imbedding theorems of Sobolev type in potential theory,, Math. Scand., 45 (1979), 77.   Google Scholar

[31]

M. Hoffman-Ostenhof, T. Hoffman-Ostenhof and A. Laptev, A geometrical version of Hardy's inequality,, J. Funct. Anal., 189 (2002), 539.   Google Scholar

[32]

B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE,", Lecture Notes in Math. 1150, (1150).   Google Scholar

[33]

V. M. Maz'ya and T. O. Shaposhnikova, "Sobolev Spaces: with Applications to Elliptic Partial Differential Equations,", Springer-Verlag, (2011).  doi: 10.1007/978-3-642-15564-2.  Google Scholar

[34]

J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (): 1077.   Google Scholar

[35]

R. O'Neil, Convolution operators in $L(p,q)$ spaces,, Duke Math. J., 30 (1963), 129.  doi: 10.1215/S0012-7094-63-03015-1.  Google Scholar

[36]

J. Peetre, Espaces d' interpolation et théorème de Soboleff,, Ann. Inst. Fourier, 16 (1966), 279.  doi: 10.5802/aif.232.  Google Scholar

[37]

S. I. Pohozaev, On the imbedding Sobolev theorem for $p=n$,, Doklady Conference, (1965), 158.   Google Scholar

[38]

G. Sinnamon and V. D. Stepanov, The weighted Hardy inequality: new proofs and the case $p=1$,, J. London Math. Soc., 54 (1996), 89.   Google Scholar

[39]

G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl., 110 (1976), 353.  doi: 10.1007/BF02418013.  Google Scholar

[40]

J. Tidblom, A Hardy inequality in the half-space,, J. Funct. Anal., 221 (2005), 482.  doi: 10.1016/j.jfa.2004.09.014.  Google Scholar

[41]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473.   Google Scholar

[42]

V. I. Yudovic, Some estimates connected with integral operators and with solutions of elliptic equations,, Dokl. Akad. Nauk SSSR, 138 (1961), 805.   Google Scholar

[43]

J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential,, J. Funct. Anal., 173 (2000), 103.  doi: 10.1006/jfan.1999.3556.  Google Scholar

show all references

References:
[1]

Adimurthi, N. Chaudhuri and M. Ramaswamy, An improved Hardy-Sobolev inequality and its applications,, Proc. Amer. Math. Soc., 130 (2002), 489.  doi: 10.1090/S0002-9939-01-06132-9.  Google Scholar

[2]

Adimurthi and M. J. Esteban, An improved Hardy-Sobolev inequality in $W^{1,p}$ and its applications to Schrödinger operators,, NoDEA Nonlinear Differential Equation Appl., 12 (2005), 243.  doi: 10.1007/S00030-005-0009-4.  Google Scholar

[3]

A. Alvino, Sulla disuguaglianza di Sobolev in spazi di Lorentz,, Boll. Un. Mat. Ital., 5 (1977), 148.   Google Scholar

[4]

A. Alvino, P.-L.Lions and G. Trombetti, On optimization problems with prescribed rearrangements,, Nonlinear Anal., 13 (1989), 185.  doi: 10.1016/0362-546X(89)90043-6.  Google Scholar

[5]

A. Alvino, R. Volpicelli and B. Volzone, On Hardy inequalities with a remainder term,, Ric. Mat., 59 (2010), 265.  doi: 10.1007/s11587-010-0086-5.  Google Scholar

[6]

T. Aubin, Problèmes isopérimetriques et espaces de Sobolev,, J. Diff. Geom., 11 (1976), 573.   Google Scholar

[7]

G. Barbatis, S. Filippas and A. Tertikas, Series expansion for $L^p$ Hardy inequalities,, Indiana Univ. Math. J., 52 (2003), 171.  doi: 0.1512/iumj.2003.52.2207.  Google Scholar

[8]

G. Barbatis, S. Filippas and A. Tertikas, A unified approach to improved $L^p$ Hardy inequalities with best constants,, Trans. Amer. Math. Soc., 356 (2004), 2169.  doi: 10.1090/S0002-9947-03-03389-0.  Google Scholar

[9]

R. Benguria, R. Frank and M. Loss, The sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half-space,, Math. Res. Lett., 15 (2008), 613.   Google Scholar

[10]

C. Bennett and R. Sharpley, "Interpolation of Operators,", Academic Press, (1988).   Google Scholar

[11]

H.Brezis and E.Lieb, Sobolev inequalities with remainder terms,, J. Funct. Anal., 62 (1985), 73.  doi: 10.1016/0022-1236(85)90020-5.  Google Scholar

[12]

H. Brezis and M. Marcus, Hardy's inequalities revisited,, Ann. Sc. Norm. Super. Pisa, 25 (1997), 217.   Google Scholar

[13]

H. Brezis, M. Marcus and I. Shafrir, Extremal functions for Hardy's inequality with weight,, J. Funct. Anal., 171 (2000), 177.  doi: 10.1006/jfan.1999.3504.  Google Scholar

[14]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437.  doi: 10.1002/cpa.3160360405.  Google Scholar

[15]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities,, Comm. Part. Diff. Eq., 5 (1980), 773.  doi: 10.1080/03605308008820154.  Google Scholar

[16]

J. E. Brothers and W. P. Ziemer, Minimal rearrangements of Sobolev functions,, J. Reine Angew. Math., 384 (1988), 419.  doi: 10.1515/crll.1988.384.153.  Google Scholar

[17]

A. Cianchi and A. Ferone, Best remainder norms in Sobolev-Hardy inequalities,, Indiana Univ. Math. J., 58 (2009), 1051.  doi: 10.1512/iumj.2009.58.3561.  Google Scholar

[18]

G. Crasta, I. Fragalà and F. Gazzola, Some estimates for the torsional rigidity of composite rods,, Math. Nachr., 280 (2007), 242.  doi: 0.1002/mana.200410478.  Google Scholar

[19]

M. Cwikel and E. Pustylnik, Sobolev type embeddings in the limiting case,, J. Fourier Anal. Appl., 4 (1998), 433.  doi: 10.1007/BF02498218.  Google Scholar

[20]

A. Detalla, T. Horiuchi and H. Ando, Missing terms in Hardy-Sobolev inequalities and its applications,, Far East J. Math. Sci., 14 (2004), 333.   Google Scholar

[21]

J. Dolbeault, M. J. Esteban, M. Loss and L. Vega, An analytical proof of Hardy-like inequalities related to the Dirac operator,, J. Funct. Anal., 216 (2004), 1.  doi: 10.1016/j.jfa.2003.09.010.  Google Scholar

[22]

D. E. Edmunds, R. A. Kerman and L. Pick, Optimal Sobolev imbeddings involving rearrangement invariant quasi-norms,, J. Funct. Anal., 170 (2000), 307.  doi: 10.1006/jfan.1999.3508.  Google Scholar

[23]

S. Filippas, V. G. Maz'ya and A. Tertikas, On a question of Brezis and Marcus,, Calc. Var. Partial Differential Equations, 25 (2006), 491.  doi: 10.1007/s00526-005-0353-6.  Google Scholar

[24]

S. Filippas, V. G. Maz'ya and A. Tertikas, Critical Hardy-Sobolev inequalities,, J. Math. Pures Appl., 87 (2007), 37.  doi: 10.1016/j.matpur.2006.10.007.  Google Scholar

[25]

F. Gazzola, H. C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms,, Trans. Amer. Math. Soc., 356 (2004), 2149.  doi: 10.1090/S0002-9947-03-03395-6.  Google Scholar

[26]

N. Ghoussoub and A. Moradifam, On the best possible remaining term in the Hardy inequality,, Proc. Natl. Acad. Sci. USA, 105 (2008), 13746.  doi: 10.1073/pnas.0803703105.  Google Scholar

[27]

E. Giarrusso and D. Nunziante, Symmetrization in a class of first-order Hamilton-Jacobi equations,, Nonlinear Anal., 8 (1984), 289.  doi: 10.1016/0362-546X(84)90031-2.  Google Scholar

[28]

A. Gogatishvili, B. Opic and L. Pick, Weighted inequalities for Hardy-type operators involving suprema,, Collect. Math., 57 (2006), 227.   Google Scholar

[29]

A. Gogatishvili and L. Pick, Discretization and antidiscretization of rearrangement-invariant norms,, Publ. Mat., 47 (2003), 311.   Google Scholar

[30]

K. Hansson, Imbedding theorems of Sobolev type in potential theory,, Math. Scand., 45 (1979), 77.   Google Scholar

[31]

M. Hoffman-Ostenhof, T. Hoffman-Ostenhof and A. Laptev, A geometrical version of Hardy's inequality,, J. Funct. Anal., 189 (2002), 539.   Google Scholar

[32]

B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE,", Lecture Notes in Math. 1150, (1150).   Google Scholar

[33]

V. M. Maz'ya and T. O. Shaposhnikova, "Sobolev Spaces: with Applications to Elliptic Partial Differential Equations,", Springer-Verlag, (2011).  doi: 10.1007/978-3-642-15564-2.  Google Scholar

[34]

J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (): 1077.   Google Scholar

[35]

R. O'Neil, Convolution operators in $L(p,q)$ spaces,, Duke Math. J., 30 (1963), 129.  doi: 10.1215/S0012-7094-63-03015-1.  Google Scholar

[36]

J. Peetre, Espaces d' interpolation et théorème de Soboleff,, Ann. Inst. Fourier, 16 (1966), 279.  doi: 10.5802/aif.232.  Google Scholar

[37]

S. I. Pohozaev, On the imbedding Sobolev theorem for $p=n$,, Doklady Conference, (1965), 158.   Google Scholar

[38]

G. Sinnamon and V. D. Stepanov, The weighted Hardy inequality: new proofs and the case $p=1$,, J. London Math. Soc., 54 (1996), 89.   Google Scholar

[39]

G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl., 110 (1976), 353.  doi: 10.1007/BF02418013.  Google Scholar

[40]

J. Tidblom, A Hardy inequality in the half-space,, J. Funct. Anal., 221 (2005), 482.  doi: 10.1016/j.jfa.2004.09.014.  Google Scholar

[41]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473.   Google Scholar

[42]

V. I. Yudovic, Some estimates connected with integral operators and with solutions of elliptic equations,, Dokl. Akad. Nauk SSSR, 138 (1961), 805.   Google Scholar

[43]

J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential,, J. Funct. Anal., 173 (2000), 103.  doi: 10.1006/jfan.1999.3556.  Google Scholar

[1]

José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138
[2]

Neal Bez, Sanghyuk Lee, Shohei Nakamura, Yoshihiro Sawano. Sharpness of the Brascamp–Lieb inequality in Lorentz spaces. Electronic Research Announcements, 2017, 24: 53-63. doi: 10.3934/era.2017.24.006
[3]

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

Jann-Long Chern, Yong-Li Tang, Chuan-Jen Chyan, Yi-Jung Chen. On the uniqueness of singular solutions for a Hardy-Sobolev equation. Conference Publications, 2013, 2013 (special) : 123-128. doi: 10.3934/proc.2013.2013.123
[5]

Paolo Maremonti. A remark on the Stokes problem in Lorentz spaces. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1323-1342. doi: 10.3934/dcdss.2013.6.1323
[6]

Masato Hashizume, Chun-Hsiung Hsia, Gyeongha Hwang. On the Neumann problem of Hardy-Sobolev critical equations with the multiple singularities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 301-322. doi: 10.3934/cpaa.2019016
[7]

Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527
[8]

Jinhui Chen, Haitao Yang. A result on Hardy-Sobolev critical elliptic equations with boundary singularities. Communications on Pure & Applied Analysis, 2007, 6 (1) : 191-201. doi: 10.3934/cpaa.2007.6.191
[9]

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
[10]

Jun Yang, Yaotian Shen. Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2565-2575. doi: 10.3934/cpaa.2013.12.2565
[11]

Haim Brezis, Petru Mironescu. Composition in fractional Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 241-246. doi: 10.3934/dcds.2001.7.241
[12]

Elvise Berchio, Debdip Ganguly. Improved higher order poincaré inequalities on the hyperbolic space via Hardy-type remainder terms. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1871-1892. doi: 10.3934/cpaa.2016020
[13]

Guoqing Zhang, Jia-yu Shao, Sanyang Liu. Linking solutions for N-laplace elliptic equations with Hardy-Sobolev operator and indefinite weights. Communications on Pure & Applied Analysis, 2011, 10 (2) : 571-581. doi: 10.3934/cpaa.2011.10.571
[14]

Wei Dai, Zhao Liu, Guozhen Lu. Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1253-1264. doi: 10.3934/cpaa.2017061
[15]

Tahar Z. Boulmezaoud, Amel Kourta. Some identities on weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 427-434. doi: 10.3934/dcdss.2012.5.427
[16]

Genggeng Huang, Congming Li, Ximing Yin. Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 935-942. doi: 10.3934/dcds.2015.35.935
[17]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597
[18]

Younghun Hong, Yannick Sire. On Fractional Schrödinger Equations in sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2265-2282. doi: 10.3934/cpaa.2015.14.2265
[19]

Daniel Guan. Classification of compact complex homogeneous spaces with invariant volumes. Electronic Research Announcements, 1997, 3: 90-92.

[20]

Daniel Guan. Classification of compact homogeneous spaces with invariant symplectic structures. Electronic Research Announcements, 1997, 3: 52-54.

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences