A variant of Poincaré's inequality

Ponce, Augusto C.

Comptes Rendus. Mathématique, Tome 337 (2003) no. 4, pp. 253-257.

Soit $\Omega \subset {ℝ}^N,\mathrm{N}⩾2$, un domaine lipschitzien borné. Étant donnée une suite de fonctions radiales positives $\left({\rho }_n\right)\subset {\mathrm{L}}^1\left({ℝ}^N\right)$ qui converge vers la masse de Dirac δ₀ on montre qu'il existe C>0 et n₀⩾1 tels que

$$\underset{\Omega }{\int }{\left|f-\underset{\Omega }{\int }f\right|}^p⩽C\underset{\Omega }{\int }\underset{\Omega }{\int }\frac{{|f\left(x\right)-f\left(y\right)|}^p}{{|x-y|}^p}{\rho }_n\left(\right|x-y\left|\right)\mathrm{d}x\mathrm{d}y\forall f\in L^p\left(\Omega \right)\forall n⩾n_0.$$

Cette estimation a été motivée par un travail récent de Bourgain, Brezis et Mironescu (dans : Optimal Control and Partial Differential Equations, IOS Press, 2001, pp. 439–455). En prenant la limite dans (2) lorsque n→∞, on retrouve l'inégalité de Poincaré. On généralise aussi un théorème de compacité de Bourgain, Brezis et Mironescu.

We show that if $\Omega \subset {ℝ}^N,\mathrm{N}⩾2$, is a bounded Lipschitz domain and $\left({\rho }_n\right)\subset {\mathrm{L}}^1\left({ℝ}^N\right)$ is a sequence of nonnegative radial functions weakly converging to δ₀ then there exist C>0 and n₀⩾1 such that

$$\underset{\Omega }{\int }{\left|f-\underset{\Omega }{\int }f\right|}^p⩽C\underset{\Omega }{\int }\underset{\Omega }{\int }\frac{{|f\left(x\right)-f\left(y\right)|}^p}{{|x-y|}^p}{\rho }_n\left(\right|x-y\left|\right)\mathrm{d}x\mathrm{d}y\forall f\in L^p\left(\Omega \right)\forall n⩾n_0.$$

The above estimate was suggested by some recent work of Bourgain, Brezis and Mironescu (in: Optimal Control and Partial Differential Equations, IOS Press, 2001, pp. 439–455). As n→∞ in (1) we recover Poincaré's inequality. We also extend a compactness result of Bourgain, Brezis and Mironescu.

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