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]

Proc. Amer. Math. Soc., 130 (2002), 489-505. doi: 10.1090/S0002-9939-01-06132-9.  Google Scholar

[2]

NoDEA Nonlinear Differential Equation Appl., 12 (2005), 243-263. doi: 10.1007/S00030-005-0009-4.  Google Scholar

[3]

Boll. Un. Mat. Ital., 5 (1977), 148-156.  Google Scholar

[4]

Nonlinear Anal., 13 (1989), 185-220. doi: 10.1016/0362-546X(89)90043-6.  Google Scholar

[5]

Ric. Mat., 59 (2010), 265-280. doi: 10.1007/s11587-010-0086-5.  Google Scholar

[6]

J. Diff. Geom., 11 (1976), 573-598.  Google Scholar

[7]

Indiana Univ. Math. J., 52 (2003), 171-190. doi: 0.1512/iumj.2003.52.2207.  Google Scholar

[8]

Trans. Amer. Math. Soc., 356 (2004), 2169-2196. doi: 10.1090/S0002-9947-03-03389-0.  Google Scholar

[9]

Math. Res. Lett., 15 (2008), 613-622.  Google Scholar

[10]

Academic Press, Boston, 1988.  Google Scholar

[11]

J. Funct. Anal., 62 (1985), 73-86. doi: 10.1016/0022-1236(85)90020-5.  Google Scholar

[12]

Ann. Sc. Norm. Super. Pisa, 25 (1997), 217-237.  Google Scholar

[13]

J. Funct. Anal., 171 (2000), 177-191. doi: 10.1006/jfan.1999.3504.  Google Scholar

[14]

Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.  Google Scholar

[15]

Comm. Part. Diff. Eq., 5 (1980), 773-789. doi: 10.1080/03605308008820154.  Google Scholar

[16]

J. Reine Angew. Math., 384 (1988), 419-431. doi: 10.1515/crll.1988.384.153.  Google Scholar

[17]

Indiana Univ. Math. J., 58 (2009), 1051-1096. doi: 10.1512/iumj.2009.58.3561.  Google Scholar

[18]

Math. Nachr., 280 (2007), 242-255. doi: 0.1002/mana.200410478.  Google Scholar

[19]

J. Fourier Anal. Appl., 4 (1998), 433-446. doi: 10.1007/BF02498218.  Google Scholar

[20]

Far East J. Math. Sci., 14 (2004), 333-359.  Google Scholar

[21]

J. Funct. Anal., 216 (2004), 1-21. doi: 10.1016/j.jfa.2003.09.010.  Google Scholar

[22]

J. Funct. Anal., 170 (2000), 307-355. doi: 10.1006/jfan.1999.3508.  Google Scholar

[23]

Calc. Var. Partial Differential Equations, 25 (2006), 491-501. doi: 10.1007/s00526-005-0353-6.  Google Scholar

[24]

J. Math. Pures Appl., 87 (2007), 37-56. doi: 10.1016/j.matpur.2006.10.007.  Google Scholar

[25]

Trans. Amer. Math. Soc., 356 (2004), 2149-2168. doi: 10.1090/S0002-9947-03-03395-6.  Google Scholar

[26]

Proc. Natl. Acad. Sci. USA, 105 (2008), 13746-13751. doi: 10.1073/pnas.0803703105.  Google Scholar

[27]

Nonlinear Anal., 8 (1984), 289-299. doi: 10.1016/0362-546X(84)90031-2.  Google Scholar

[28]

Collect. Math., 57 (2006), 227-255.  Google Scholar

[29]

Publ. Mat., 47 (2003), 311-358.  Google Scholar

[30]

Math. Scand., 45 (1979), 77-102.  Google Scholar

[31]

J. Funct. Anal., 189 (2002), 539-548.  Google Scholar

[32]

Lecture Notes in Math. 1150, Springer-Verlag, Berlin-New York, 1985.  Google Scholar

[33]

Springer-Verlag, Berlin Heidelberg, 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]

Duke Math. J., 30 (1963), 129-142. doi: 10.1215/S0012-7094-63-03015-1.  Google Scholar

[36]

Ann. Inst. Fourier, 16 (1966), 279-317. doi: 10.5802/aif.232.  Google Scholar

[37]

Doklady Conference, Section Math. Moscow Power Inst., (1965), 158-170 (Russian). Google Scholar

[38]

J. London Math. Soc., 54 (1996), 89-101.  Google Scholar

[39]

Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.  Google Scholar

[40]

J. Funct. Anal., 221 (2005), 482-495. doi: 10.1016/j.jfa.2004.09.014.  Google Scholar

[41]

J. Math. Mech., 17 (1967), 473-483.  Google Scholar

[42]

Dokl. Akad. Nauk SSSR, 138 (1961), 805-808 (Russian);  Google Scholar

[43]

J. Funct. Anal., 173 (2000), 103-153. doi: 10.1006/jfan.1999.3556.  Google Scholar

show all references

References:
[1]

Proc. Amer. Math. Soc., 130 (2002), 489-505. doi: 10.1090/S0002-9939-01-06132-9.  Google Scholar

[2]

NoDEA Nonlinear Differential Equation Appl., 12 (2005), 243-263. doi: 10.1007/S00030-005-0009-4.  Google Scholar

[3]

Boll. Un. Mat. Ital., 5 (1977), 148-156.  Google Scholar

[4]

Nonlinear Anal., 13 (1989), 185-220. doi: 10.1016/0362-546X(89)90043-6.  Google Scholar

[5]

Ric. Mat., 59 (2010), 265-280. doi: 10.1007/s11587-010-0086-5.  Google Scholar

[6]

J. Diff. Geom., 11 (1976), 573-598.  Google Scholar

[7]

Indiana Univ. Math. J., 52 (2003), 171-190. doi: 0.1512/iumj.2003.52.2207.  Google Scholar

[8]

Trans. Amer. Math. Soc., 356 (2004), 2169-2196. doi: 10.1090/S0002-9947-03-03389-0.  Google Scholar

[9]

Math. Res. Lett., 15 (2008), 613-622.  Google Scholar

[10]

Academic Press, Boston, 1988.  Google Scholar

[11]

J. Funct. Anal., 62 (1985), 73-86. doi: 10.1016/0022-1236(85)90020-5.  Google Scholar

[12]

Ann. Sc. Norm. Super. Pisa, 25 (1997), 217-237.  Google Scholar

[13]

J. Funct. Anal., 171 (2000), 177-191. doi: 10.1006/jfan.1999.3504.  Google Scholar

[14]

Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.  Google Scholar

[15]

Comm. Part. Diff. Eq., 5 (1980), 773-789. doi: 10.1080/03605308008820154.  Google Scholar

[16]

J. Reine Angew. Math., 384 (1988), 419-431. doi: 10.1515/crll.1988.384.153.  Google Scholar

[17]

Indiana Univ. Math. J., 58 (2009), 1051-1096. doi: 10.1512/iumj.2009.58.3561.  Google Scholar

[18]

Math. Nachr., 280 (2007), 242-255. doi: 0.1002/mana.200410478.  Google Scholar

[19]

J. Fourier Anal. Appl., 4 (1998), 433-446. doi: 10.1007/BF02498218.  Google Scholar

[20]

Far East J. Math. Sci., 14 (2004), 333-359.  Google Scholar

[21]

J. Funct. Anal., 216 (2004), 1-21. doi: 10.1016/j.jfa.2003.09.010.  Google Scholar

[22]

J. Funct. Anal., 170 (2000), 307-355. doi: 10.1006/jfan.1999.3508.  Google Scholar

[23]

Calc. Var. Partial Differential Equations, 25 (2006), 491-501. doi: 10.1007/s00526-005-0353-6.  Google Scholar

[24]

J. Math. Pures Appl., 87 (2007), 37-56. doi: 10.1016/j.matpur.2006.10.007.  Google Scholar

[25]

Trans. Amer. Math. Soc., 356 (2004), 2149-2168. doi: 10.1090/S0002-9947-03-03395-6.  Google Scholar

[26]

Proc. Natl. Acad. Sci. USA, 105 (2008), 13746-13751. doi: 10.1073/pnas.0803703105.  Google Scholar

[27]

Nonlinear Anal., 8 (1984), 289-299. doi: 10.1016/0362-546X(84)90031-2.  Google Scholar

[28]

Collect. Math., 57 (2006), 227-255.  Google Scholar

[29]

Publ. Mat., 47 (2003), 311-358.  Google Scholar

[30]

Math. Scand., 45 (1979), 77-102.  Google Scholar

[31]

J. Funct. Anal., 189 (2002), 539-548.  Google Scholar

[32]

Lecture Notes in Math. 1150, Springer-Verlag, Berlin-New York, 1985.  Google Scholar

[33]

Springer-Verlag, Berlin Heidelberg, 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]

Duke Math. J., 30 (1963), 129-142. doi: 10.1215/S0012-7094-63-03015-1.  Google Scholar

[36]

Ann. Inst. Fourier, 16 (1966), 279-317. doi: 10.5802/aif.232.  Google Scholar

[37]

Doklady Conference, Section Math. Moscow Power Inst., (1965), 158-170 (Russian). Google Scholar

[38]

J. London Math. Soc., 54 (1996), 89-101.  Google Scholar

[39]

Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.  Google Scholar

[40]

J. Funct. Anal., 221 (2005), 482-495. doi: 10.1016/j.jfa.2004.09.014.  Google Scholar

[41]

J. Math. Mech., 17 (1967), 473-483.  Google Scholar

[42]

Dokl. Akad. Nauk SSSR, 138 (1961), 805-808 (Russian);  Google Scholar

[43]

J. Funct. Anal., 173 (2000), 103-153. doi: 10.1006/jfan.1999.3556.  Google Scholar

[1]

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
[2]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115
[3]

Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511
[4]

Fabio Sperotto Bemfica, Marcelo Mendes Disconzi, Casey Rodriguez, Yuanzhen Shao. Local existence and uniqueness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021069
[5]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037
[6]

Xiaorong Luo, Anmin Mao, Yanbin Sang. Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021022
[7]

Krzysztof A. Krakowski, Luís Machado, Fátima Silva Leite. A unifying approach for rolling symmetric spaces. Journal of Geometric Mechanics, 2021, 13 (1) : 145-166. doi: 10.3934/jgm.2020016
[8]

Zhengchao Ji. Cylindrical estimates for mean curvature flow in hyperbolic spaces. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1199-1211. doi: 10.3934/cpaa.2021016
[9]

Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119
[10]

Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065
[11]

Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934/dcds.2020393
[12]

K. Ravikumar, Manil T. Mohan, A. Anguraj. Approximate controllability of a non-autonomous evolution equation in Banach spaces. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 461-485. doi: 10.3934/naco.2020038
[13]

Mao Okada. Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces. Journal of Modern Dynamics, 2021, 17: 111-143. doi: 10.3934/jmd.2021004
[14]

Jinyi Sun, Zunwei Fu, Yue Yin, Minghua Yang. Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3409-3425. doi: 10.3934/dcdsb.2020237
[15]

Tôn Việt Tạ. Strict solutions to stochastic semilinear evolution equations in M-type 2 Banach spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021050
[16]

Kazeem Olalekan Aremu, Chinedu Izuchukwu, Grace Nnenanya Ogwo, Oluwatosin Temitope Mewomo. Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2161-2180. doi: 10.3934/jimo.2020063
[17]

Khalid Latrach, Hssaine Oummi, Ahmed Zeghal. Existence results for nonlinear mono-energetic singular transport equations: $ L^p $-spaces. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021028
[18]

Andreia Chapouto. A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3915-3950. doi: 10.3934/dcds.2021022
[19]

José Antonio Carrillo, Martin Parisot, Zuzanna Szymańska. Mathematical modelling of collagen fibres rearrangement during the tendon healing process. Kinetic & Related Models, 2021, 14 (2) : 283-301. doi: 10.3934/krm.2021005
[20]

Andreas Neubauer. On Tikhonov-type regularization with approximated penalty terms. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021027

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences