A sharp inequality for Sobolev functions
Comptes Rendus. Mathématique, Volume 334 (2002) no. 2, pp. 105-108.

Let N⩾5, a>0, $\Omega$ be a smooth bounded domain in ${ℝ}^{N}$, ${2}^{*}=\frac{2N}{N-2}$, ${2}^{#}=\frac{2\left(\mathrm{N}-1\right)}{\mathrm{N}-2}$ and ‖u2=|∇u|22+a|u|22. We prove there exists an α0>0 such that, for all $\mathrm{u}\in {\mathrm{H}}^{1}\left(\Omega \right)⧹\left\{0\right\}$,

 $\frac{S}{{2}^{2/N}}⩽\frac{{\parallel u\parallel }^{2}}{{|u|}_{{2}^{*}}^{2}}\left(1+{\alpha }_{0}\frac{{|u|}_{{2}^{#}}^{{2}^{#}}}{{\parallel u\parallel ·|u|}_{{2}^{*}}^{{2}^{*}/2}}\right).$
This inequality implies Cherrier's inequality.

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

 $\frac{S}{{2}^{2/N}}⩽\frac{{\parallel u\parallel }^{2}}{{|u|}_{{2}^{*}}^{2}}\left(1+{\alpha }_{0}\frac{{|u|}_{{2}^{#}}^{{2}^{#}}}{{\parallel u\parallel ·|u|}_{{2}^{*}}^{{2}^{*}/2}}\right).$
Cette inégalité implique l'inégalité de Cherrier.

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

Pedro M. Girão 1

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

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