Sharp Hardy–Sobolev inequalities

S. Filippas; V.G. Maz'ya; A. Tertikas

doi:10.1016/j.crma.2004.07.023

Partial Differential Equations

Sharp Hardy–Sobolev inequalities

Presented by: Haïm Brezis

S. Filippas ^1,²; V.G. Maz'ya ^3,⁴; A. Tertikas ^2,⁵

¹ Department of Applied Mathematics, University of Crete, 71409 Heraklion, Greece
² Institute of Applied and Computational Mathematics FORTH, 71110 Heraklion, Greece
³ Department of Mathematics, Ohio State University, Columbus, OH 43210, USA
⁴ Department of Mathematics, Linkoeping University, 58183 Linkoeping, Sweden
⁵ Department of Mathematics, University of Crete, 71409 Heraklion, Greece

Comptes Rendus. Mathématique, Volume 339 (2004) no. 7, pp. 483-486.

Abstract
Résumé

Let Ω be a smooth bounded domain in $R^{N}$ , $N ⩾ 3$ . We show that Hardy's inequality involving the distance to the boundary, with best constant ( $\frac{1}{4}$ ), may still be improved by adding a multiple of the critical Sobolev norm.

Soit Ω un ouvert borné et regulier dans $R^{N}$ , $N ⩾ 3$ . On montre que l'inegalité de Hardy, liée à la distance au bord, avec meilleure constante ( $\frac{1}{4}$ ), peut être améliorée en ajoutant un multiple de la norme de Sobolev critique.

Received: 2004-07-26
Published online: 2004-09-25

DOI: 10.1016/j.crma.2004.07.023

Author's affiliations:

S. Filippas ^{1, 2}; V.G. Maz'ya ^{3, 4}; A. Tertikas ^{2, 5}

¹ Department of Applied Mathematics, University of Crete, 71409 Heraklion, Greece
² Institute of Applied and Computational Mathematics FORTH, 71110 Heraklion, Greece
³ Department of Mathematics, Ohio State University, Columbus, OH 43210, USA
⁴ Department of Mathematics, Linkoeping University, 58183 Linkoeping, Sweden
⁵ Department of Mathematics, University of Crete, 71409 Heraklion, Greece

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     author = {S. Filippas and V.G. Maz'ya and A. Tertikas},
     title = {Sharp {Hardy{\textendash}Sobolev} inequalities},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {483--486},
     publisher = {Elsevier},
     volume = {339},
     number = {7},
     year = {2004},
     doi = {10.1016/j.crma.2004.07.023},
     language = {en},
}

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S. Filippas; V.G. Maz'ya; A. Tertikas. Sharp Hardy–Sobolev inequalities. Comptes Rendus. Mathématique, Volume 339 (2004) no. 7, pp. 483-486. doi : 10.1016/j.crma.2004.07.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.07.023/

References
Cited by

[1] G. Barbatis; S. Filippas; A. Tertikas A unified approach to improved $L^{p}$ Hardy inequalities with best constants, Trans. Amer. Math. Soc., Volume 356 (2004) no. 6, pp. 2169-2196

[2] H. Brezis; M. Marcus Hardy's inequalities revisited, Ann. Scuola Norm. Pisa, Volume 25 (1997), pp. 217-237

[3] J. Dávila; L. Dupaigne Hardy-type inequalities, J. Eur. Math. Soc., Volume 6 (2004) no. 3, pp. 335-365

[4] S. Filippas, V.G. Maz'ya, A. Tertikas, Critical Hardy Sobolev inequalities, in preparation

[5] S. Filippas; A. Tertikas Optimizing improved Hardy inequalities, J. Funct. Anal., Volume 192 (2002), pp. 186-233

[6] V.G. Maz'ya Sobolev Spaces, Springer, 1985

[7] J.L. Vázquez; E. Zuazua The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., Volume 173 (2000), pp. 103-153

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