A sharp inequality for Sobolev functions

Girão, Pedro M.

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

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

$$\frac{S}{2^{2/N}}⩽\frac{{\parallel u\parallel }^2}{{\left|u\right|}_{2^{*}}^2}\left(1+{\alpha }_0\frac{{\left|u\right|}_{2^{\#}}^{2^{\#}}}{{\parallel u\parallel ·\left|u\right|}_{2^{*}}^{2^{*}/2}}\right).$$

Cette inégalité implique l'inégalité de Cherrier.

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

$$\frac{S}{2^{2/N}}⩽\frac{{\parallel u\parallel }^2}{{\left|u\right|}_{2^{*}}^2}\left(1+{\alpha }_0\frac{{\left|u\right|}_{2^{\#}}^{2^{\#}}}{{\parallel u\parallel ·\left|u\right|}_{2^{*}}^{2^{*}/2}}\right).$$

This inequality implies Cherrier's inequality.

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