A sharp inequality for Sobolev functions

Pedro M. Girão

doi:10.1016/S1631-073X(02)02215-X

A sharp inequality for Sobolev functions
[Une inégalité dans un espace de Sobolev]

Pedro M. Girão ¹

¹ Department of Mathematics, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

Comptes Rendus. Mathématique, Volume 334 (2002) no. 2, pp. 105-108.

Résumé
Abstract

Nous considérons N⩾5, a>0, $Ω$ un ouvert borné régulier de $ℝ^{N}$ , $2^{*} = \frac{2 N}{N - 2}$ , $2^{#} = \frac{2 (N - 1)}{N - 2}$ et ||u||²=|∇u|₂²+a|u|₂². Nous prouvons qu'il existe α₀>0 tel que, pour toute fonction $u \in H^{1} (Ω) ⧹ {0}$ ,

\frac{S}{2^{2 / N}} ⩽ \frac{{∥ u ∥}^{2}}{{| u |}_{2^{*}}^{2}} (1 + α_{0} \frac{{| u |}_{2^{#}}^{2^{#}}}{{∥ u ∥ \cdot | u |}_{2^{*}}^{2^{*} / 2}}) .

Cette inégalité implique l'inégalité de Cherrier.

Let N⩾5, a>0, $Ω$ be a smooth bounded domain in $ℝ^{N}$ , $2^{*} = \frac{2 N}{N - 2}$ , $2^{#} = \frac{2 (N - 1)}{N - 2}$ and ‖u‖²=|∇u|₂²+a|u|₂². We prove there exists an α₀>0 such that, for all $u \in H^{1} (Ω) ⧹ {0}$ ,

\frac{S}{2^{2 / N}} ⩽ \frac{{∥ u ∥}^{2}}{{| u |}_{2^{*}}^{2}} (1 + α_{0} \frac{{| u |}_{2^{#}}^{2^{#}}}{{∥ u ∥ \cdot | u |}_{2^{*}}^{2^{*} / 2}}) .

This inequality implies Cherrier's inequality.

Accepté le : 2001-11-15
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02215-X

Affiliations des auteurs :

Pedro M. Girão ¹

¹ Department of Mathematics, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

@article{CRMATH_2002__334_2_105_0,
     author = {Pedro M. Gir\~ao},
     title = {A sharp inequality for {Sobolev} functions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {105--108},
     publisher = {Elsevier},
     volume = {334},
     number = {2},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02215-X},
     language = {en},
}

TY  - JOUR
AU  - Pedro M. Girão
TI  - A sharp inequality for Sobolev functions
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 105
EP  - 108
VL  - 334
IS  - 2
PB  - Elsevier
DO  - 10.1016/S1631-073X(02)02215-X
LA  - en
ID  - CRMATH_2002__334_2_105_0
ER  -

%0 Journal Article
%A Pedro M. Girão
%T A sharp inequality for Sobolev functions
%J Comptes Rendus. Mathématique
%D 2002
%P 105-108
%V 334
%N 2
%I Elsevier
%R 10.1016/S1631-073X(02)02215-X
%G en
%F CRMATH_2002__334_2_105_0

Pedro M. Girão. A sharp inequality for Sobolev functions. Comptes Rendus. Mathématique, Volume 334 (2002) no. 2, pp. 105-108. doi : 10.1016/S1631-073X(02)02215-X. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02215-X/

Bibliographie
Cité par

[1] Adimurthi; G. Mancini The Neumann problem for elliptic equations with critical nonlinearity, A tribute in honour of G. Prodi, Scuola Norm. Sup. Pisa (1991), pp. 9-25

[2] Adimurthi; F. Pacella; S.L. Yadava Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity, J. Funct. Anal., Volume 113 (1993), pp. 318-350

[3] H. Brezis; E. Lieb Sobolev inequalities with remainder terms, J. Funct. Anal., Volume 62 (1985), pp. 73-86

[4] H. Brezis; L. Nirenberg Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., Volume 36 (1983), pp. 437-477

[5] Chabrowski J., Willem M., Least energy solutions for the critical Neumann problem with a weight, Calc. Var. (to appear)

[6] P. Cherrier Problémes de Neumann nonlinéaires sur des variétés Riemannienes, J. Funct. Anal., Volume 57 (1984), pp. 154-207

[7] Costa D.G., Girão P.M., Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbation (to appear)

[8] P.-L. Lions The concentration-compactness principle in the calculus of variations, The limit case, Rev. Math. Iberoamericana, Volume 1 (1985) no. 1–2, pp. 145-201 (and 45–120)

[9] X.J. Wang Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations, Volume 93 (1991), pp. 283-310

[10] Z.Q. Wang Existence and nonexistence of G-least energy solutions for a nonlinear Neumann problem with critical exponent in symmetric domains, Calc. Var., Volume 8 (1999), pp. 109-122

[11] M. Zhu Sharp Sobolev inequalities with interior norms, Calc. Var., Volume 8 (1999), pp. 27-43

Cité par Sources :

Commentaires - Politique