A variant of Poincaré's inequality

Ponce, Augusto C.

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

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.

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.

[1] J. Bourgain, H. Brezis, personal communication

[2] Bourgain, J.; Brezis, H.; Mironescu, P. Another look at Sobolev spaces (Menaldi, J.L.; Rofman, E.; Sulem, A., eds.), Optimal Control and Partial Differential Equations, IOS Press, 2001, pp. 439-455 (A volume in honour of A. Benssoussan's 60th birthday)

[3] Bourgain, J.; Brezis, H.; Mironescu, P. Limiting embedding theorems for W^s,p when s↑1 and applications, J. Anal. Math., Volume 87 (2002), pp. 77-101 (Dedicated to the memory of Thomas H. Wolff)

[4] J. Bourgain, H. Brezis, P. Mironescu, H^1/2 maps with values into the circle: minimal connections, lifting, and the Ginzburg–Landau equation, in press

[5] Brezis, H. How to recognize constant functions. Connections with Sobolev spaces, Uspekhi Mat. Nauk, Volume 57 (2002), pp. 59-74 (in Russian). English version: Russian Math. Surveys, 57, 2002, pp. 693-708 Volume in honor of M. Vishik

[6] Maz'ya, V.; Shaposhnikova, T. On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., Volume 195 (2002), pp. 230-238 (Erratum J. Funct. Anal., 201, 2003, pp. 298-300)

[7] A.C. Ponce, An estimate in the spirit of Poincaré's inequality, J. Eur. Math. Soc., in press