American Institute of Mathematical Sciences

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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
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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]

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[29]

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

[30]

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[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

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