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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 and 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), 489-505. doi: 10.1090/S0002-9939-01-06132-9.

[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), 243-263. doi: 10.1007/S00030-005-0009-4.

[3]

A. Alvino, Sulla disuguaglianza di Sobolev in spazi di Lorentz, Boll. Un. Mat. Ital., 5 (1977), 148-156.

[4]

A. Alvino, P.-L.Lions and G. Trombetti, On optimization problems with prescribed rearrangements, Nonlinear Anal., 13 (1989), 185-220. doi: 10.1016/0362-546X(89)90043-6.

[5]

A. Alvino, R. Volpicelli and B. Volzone, On Hardy inequalities with a remainder term, Ric. Mat., 59 (2010), 265-280. doi: 10.1007/s11587-010-0086-5.

[6]

T. Aubin, Problèmes isopérimetriques et espaces de Sobolev, J. Diff. Geom., 11 (1976), 573-598.

[7]

G. Barbatis, S. Filippas and A. Tertikas, Series expansion for $L^p$ Hardy inequalities, Indiana Univ. Math. J., 52 (2003), 171-190. doi: 0.1512/iumj.2003.52.2207.

[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), 2169-2196. doi: 10.1090/S0002-9947-03-03389-0.

[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), 613-622.

[10]

C. Bennett and R. Sharpley, "Interpolation of Operators," Academic Press, Boston, 1988.

[11]

H.Brezis and E.Lieb, Sobolev inequalities with remainder terms, J. Funct. Anal., 62 (1985), 73-86. doi: 10.1016/0022-1236(85)90020-5.

[12]

H. Brezis and M. Marcus, Hardy's inequalities revisited, Ann. Sc. Norm. Super. Pisa, 25 (1997), 217-237.

[13]

H. Brezis, M. Marcus and I. Shafrir, Extremal functions for Hardy's inequality with weight, J. Funct. Anal., 171 (2000), 177-191. doi: 10.1006/jfan.1999.3504.

[14]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.

[15]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Part. Diff. Eq., 5 (1980), 773-789. doi: 10.1080/03605308008820154.

[16]

J. E. Brothers and W. P. Ziemer, Minimal rearrangements of Sobolev functions, J. Reine Angew. Math., 384 (1988), 419-431. doi: 10.1515/crll.1988.384.153.

[17]

A. Cianchi and A. Ferone, Best remainder norms in Sobolev-Hardy inequalities, Indiana Univ. Math. J., 58 (2009), 1051-1096. doi: 10.1512/iumj.2009.58.3561.

[18]

G. Crasta, I. Fragalà and F. Gazzola, Some estimates for the torsional rigidity of composite rods, Math. Nachr., 280 (2007), 242-255. doi: 0.1002/mana.200410478.

[19]

M. Cwikel and E. Pustylnik, Sobolev type embeddings in the limiting case, J. Fourier Anal. Appl., 4 (1998), 433-446. doi: 10.1007/BF02498218.

[20]

A. Detalla, T. Horiuchi and H. Ando, Missing terms in Hardy-Sobolev inequalities and its applications, Far East J. Math. Sci., 14 (2004), 333-359.

[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), 1-21. doi: 10.1016/j.jfa.2003.09.010.

[22]

D. E. Edmunds, R. A. Kerman and L. Pick, Optimal Sobolev imbeddings involving rearrangement invariant quasi-norms, J. Funct. Anal., 170 (2000), 307-355. doi: 10.1006/jfan.1999.3508.

[23]

S. Filippas, V. G. Maz'ya and A. Tertikas, On a question of Brezis and Marcus, Calc. Var. Partial Differential Equations, 25 (2006), 491-501. doi: 10.1007/s00526-005-0353-6.

[24]

S. Filippas, V. G. Maz'ya and A. Tertikas, Critical Hardy-Sobolev inequalities, J. Math. Pures Appl., 87 (2007), 37-56. doi: 10.1016/j.matpur.2006.10.007.

[25]

F. Gazzola, H. C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms, Trans. Amer. Math. Soc., 356 (2004), 2149-2168. doi: 10.1090/S0002-9947-03-03395-6.

[26]

N. Ghoussoub and A. Moradifam, On the best possible remaining term in the Hardy inequality, Proc. Natl. Acad. Sci. USA, 105 (2008), 13746-13751. doi: 10.1073/pnas.0803703105.

[27]

E. Giarrusso and D. Nunziante, Symmetrization in a class of first-order Hamilton-Jacobi equations, Nonlinear Anal., 8 (1984), 289-299. doi: 10.1016/0362-546X(84)90031-2.

[28]

A. Gogatishvili, B. Opic and L. Pick, Weighted inequalities for Hardy-type operators involving suprema, Collect. Math., 57 (2006), 227-255.

[29]

A. Gogatishvili and L. Pick, Discretization and antidiscretization of rearrangement-invariant norms, Publ. Mat., 47 (2003), 311-358.

[30]

K. Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand., 45 (1979), 77-102.

[31]

M. Hoffman-Ostenhof, T. Hoffman-Ostenhof and A. Laptev, A geometrical version of Hardy's inequality, J. Funct. Anal., 189 (2002), 539-548.

[32]

B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE," Lecture Notes in Math. 1150, Springer-Verlag, Berlin-New York, 1985.

[33]

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

[34]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.

[35]

R. O'Neil, Convolution operators in $L(p,q)$ spaces, Duke Math. J., 30 (1963), 129-142. doi: 10.1215/S0012-7094-63-03015-1.

[36]

J. Peetre, Espaces d' interpolation et théorème de Soboleff, Ann. Inst. Fourier, 16 (1966), 279-317. doi: 10.5802/aif.232.

[37]

S. I. Pohozaev, On the imbedding Sobolev theorem for $p=n$, Doklady Conference, Section Math. Moscow Power Inst., (1965), 158-170 (Russian).

[38]

G. Sinnamon and V. D. Stepanov, The weighted Hardy inequality: new proofs and the case $p=1$, J. London Math. Soc., 54 (1996), 89-101.

[39]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.

[40]

J. Tidblom, A Hardy inequality in the half-space, J. Funct. Anal., 221 (2005), 482-495. doi: 10.1016/j.jfa.2004.09.014.

[41]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.

[42]

V. I. Yudovic, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk SSSR, 138 (1961), 805-808 (Russian);

[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), 103-153. doi: 10.1006/jfan.1999.3556.

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), 489-505. doi: 10.1090/S0002-9939-01-06132-9.

[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), 243-263. doi: 10.1007/S00030-005-0009-4.

[3]

A. Alvino, Sulla disuguaglianza di Sobolev in spazi di Lorentz, Boll. Un. Mat. Ital., 5 (1977), 148-156.

[4]

A. Alvino, P.-L.Lions and G. Trombetti, On optimization problems with prescribed rearrangements, Nonlinear Anal., 13 (1989), 185-220. doi: 10.1016/0362-546X(89)90043-6.

[5]

A. Alvino, R. Volpicelli and B. Volzone, On Hardy inequalities with a remainder term, Ric. Mat., 59 (2010), 265-280. doi: 10.1007/s11587-010-0086-5.

[6]

T. Aubin, Problèmes isopérimetriques et espaces de Sobolev, J. Diff. Geom., 11 (1976), 573-598.

[7]

G. Barbatis, S. Filippas and A. Tertikas, Series expansion for $L^p$ Hardy inequalities, Indiana Univ. Math. J., 52 (2003), 171-190. doi: 0.1512/iumj.2003.52.2207.

[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), 2169-2196. doi: 10.1090/S0002-9947-03-03389-0.

[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), 613-622.

[10]

C. Bennett and R. Sharpley, "Interpolation of Operators," Academic Press, Boston, 1988.

[11]

H.Brezis and E.Lieb, Sobolev inequalities with remainder terms, J. Funct. Anal., 62 (1985), 73-86. doi: 10.1016/0022-1236(85)90020-5.

[12]

H. Brezis and M. Marcus, Hardy's inequalities revisited, Ann. Sc. Norm. Super. Pisa, 25 (1997), 217-237.

[13]

H. Brezis, M. Marcus and I. Shafrir, Extremal functions for Hardy's inequality with weight, J. Funct. Anal., 171 (2000), 177-191. doi: 10.1006/jfan.1999.3504.

[14]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.

[15]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Part. Diff. Eq., 5 (1980), 773-789. doi: 10.1080/03605308008820154.

[16]

J. E. Brothers and W. P. Ziemer, Minimal rearrangements of Sobolev functions, J. Reine Angew. Math., 384 (1988), 419-431. doi: 10.1515/crll.1988.384.153.

[17]

A. Cianchi and A. Ferone, Best remainder norms in Sobolev-Hardy inequalities, Indiana Univ. Math. J., 58 (2009), 1051-1096. doi: 10.1512/iumj.2009.58.3561.

[18]

G. Crasta, I. Fragalà and F. Gazzola, Some estimates for the torsional rigidity of composite rods, Math. Nachr., 280 (2007), 242-255. doi: 0.1002/mana.200410478.

[19]

M. Cwikel and E. Pustylnik, Sobolev type embeddings in the limiting case, J. Fourier Anal. Appl., 4 (1998), 433-446. doi: 10.1007/BF02498218.

[20]

A. Detalla, T. Horiuchi and H. Ando, Missing terms in Hardy-Sobolev inequalities and its applications, Far East J. Math. Sci., 14 (2004), 333-359.

[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), 1-21. doi: 10.1016/j.jfa.2003.09.010.

[22]

D. E. Edmunds, R. A. Kerman and L. Pick, Optimal Sobolev imbeddings involving rearrangement invariant quasi-norms, J. Funct. Anal., 170 (2000), 307-355. doi: 10.1006/jfan.1999.3508.

[23]

S. Filippas, V. G. Maz'ya and A. Tertikas, On a question of Brezis and Marcus, Calc. Var. Partial Differential Equations, 25 (2006), 491-501. doi: 10.1007/s00526-005-0353-6.

[24]

S. Filippas, V. G. Maz'ya and A. Tertikas, Critical Hardy-Sobolev inequalities, J. Math. Pures Appl., 87 (2007), 37-56. doi: 10.1016/j.matpur.2006.10.007.

[25]

F. Gazzola, H. C. Grunau and E. Mitidieri, Hardy inequalities with optimal constants and remainder terms, Trans. Amer. Math. Soc., 356 (2004), 2149-2168. doi: 10.1090/S0002-9947-03-03395-6.

[26]

N. Ghoussoub and A. Moradifam, On the best possible remaining term in the Hardy inequality, Proc. Natl. Acad. Sci. USA, 105 (2008), 13746-13751. doi: 10.1073/pnas.0803703105.

[27]

E. Giarrusso and D. Nunziante, Symmetrization in a class of first-order Hamilton-Jacobi equations, Nonlinear Anal., 8 (1984), 289-299. doi: 10.1016/0362-546X(84)90031-2.

[28]

A. Gogatishvili, B. Opic and L. Pick, Weighted inequalities for Hardy-type operators involving suprema, Collect. Math., 57 (2006), 227-255.

[29]

A. Gogatishvili and L. Pick, Discretization and antidiscretization of rearrangement-invariant norms, Publ. Mat., 47 (2003), 311-358.

[30]

K. Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand., 45 (1979), 77-102.

[31]

M. Hoffman-Ostenhof, T. Hoffman-Ostenhof and A. Laptev, A geometrical version of Hardy's inequality, J. Funct. Anal., 189 (2002), 539-548.

[32]

B. Kawohl, "Rearrangements and Convexity of Level Sets in PDE," Lecture Notes in Math. 1150, Springer-Verlag, Berlin-New York, 1985.

[33]

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

[34]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.

[35]

R. O'Neil, Convolution operators in $L(p,q)$ spaces, Duke Math. J., 30 (1963), 129-142. doi: 10.1215/S0012-7094-63-03015-1.

[36]

J. Peetre, Espaces d' interpolation et théorème de Soboleff, Ann. Inst. Fourier, 16 (1966), 279-317. doi: 10.5802/aif.232.

[37]

S. I. Pohozaev, On the imbedding Sobolev theorem for $p=n$, Doklady Conference, Section Math. Moscow Power Inst., (1965), 158-170 (Russian).

[38]

G. Sinnamon and V. D. Stepanov, The weighted Hardy inequality: new proofs and the case $p=1$, J. London Math. Soc., 54 (1996), 89-101.

[39]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.

[40]

J. Tidblom, A Hardy inequality in the half-space, J. Funct. Anal., 221 (2005), 482-495. doi: 10.1016/j.jfa.2004.09.014.

[41]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.

[42]

V. I. Yudovic, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk SSSR, 138 (1961), 805-808 (Russian);

[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), 103-153. doi: 10.1006/jfan.1999.3556.

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