A variant of Poincaré's inequality

Augusto C. Ponce

doi:10.1016/S1631-073X(03)00313-3

Partial Differential Equations

A variant of Poincaré's inequality
[Une variante de l'inégalité de Poincaré]

Présenté par : Haïm Brézis

Augusto C. Ponce ^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

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

Résumé
Abstract

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

\int_{Ω} {|f - \int_{Ω} f|}^{p} ⩽ C \int_{Ω} \int_{Ω} \frac{{| f (x) - f (y) |}^{p}}{{| x - y |}^{p}} ρ_{n} (| x - y |) d x d y \forall f \in L^{p} (Ω) \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 $Ω \subset ℝ^{N}, N ⩾ 2$ , is a bounded Lipschitz domain and $(ρ_{n}) \subset L^{1} (ℝ^{N})$ is a sequence of nonnegative radial functions weakly converging to δ₀ then there exist C>0 and n₀⩾1 such that

\int_{Ω} {|f - \int_{Ω} f|}^{p} ⩽ C \int_{Ω} \int_{Ω} \frac{{| f (x) - f (y) |}^{p}}{{| x - y |}^{p}} ρ_{n} (| x - y |) d x d y \forall f \in L^{p} (Ω) \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.

Reçu le : 2003-06-20
Accepté le : 2003-06-20
Publié le : 2003-07-29

DOI : 10.1016/S1631-073X(03)00313-3

Affiliations des auteurs :

Augusto C. Ponce ^{1, 2}

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     author = {Augusto C. Ponce},
     title = {A variant of {Poincar\'e's} inequality},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {253--257},
     publisher = {Elsevier},
     volume = {337},
     number = {4},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00313-3},
     language = {en},
}

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Augusto C. Ponce. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00313-3/

Bibliographie
Cité par

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

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[3] J. Bourgain; H. Brezis; P. Mironescu 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] H. Brezis 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] V. Maz'ya; T. Shaposhnikova 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

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