Partial Differential Equations
A variant of Poincaré's inequality
Comptes Rendus. Mathématique, Volume 337 (2003) no. 4, pp. 253-257.

We show that if Ω N ,N2, is a bounded Lipschitz domain and (ρ n )L 1 ( N ) is a sequence of nonnegative radial functions weakly converging to δ0 then there exist C>0 and n0⩾1 such that

Ωf- Ωf p C Ω Ω|f(x)-f(y)| p |x-y| p ρ n (|x-y|)dxdyfL p (Ω)nn 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 Ω N ,N2, un domaine lipschitzien borné. Étant donnée une suite de fonctions radiales positives (ρ n )L 1 ( N ) qui converge vers la masse de Dirac δ0 on montre qu'il existe C>0 et n0⩾1 tels que

Ωf- Ωf p C Ω Ω|f(x)-f(y)| p |x-y| p ρ n (|x-y|)dxdyfL p (Ω)nn 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.

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Accepted:
Published online:
DOI: 10.1016/S1631-073X(03)00313-3
Ponce, Augusto C. 1, 2

1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, BC 187, 4, pl. Jussieu, 75252 Paris cedex 05, France
2 Rutgers University, Dept. of Math., Hill Center, Busch Campus, 110 Frelinghuysen Rd, Piscataway, NJ 08854, USA
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Ponce, Augusto C. A variant of Poincaré's inequality. Comptes Rendus. Mathématique, Volume 337 (2003) no. 4, pp. 253-257. doi : 10.1016/S1631-073X(03)00313-3. http://www.numdam.org/articles/10.1016/S1631-073X(03)00313-3/

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

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