A variant of Poincaré’s inequality. (English. Abridged French version) Zbl 1073.46028
Summary: We show that if \(\Omega\subset\mathbb R^N\), \(N\geq 2\), is a bounded Lipschitz domain and \((\rho_n)\subset L^1(\mathbb R^N)\) is a sequence of nonnegative radial functions weakly converging to \(\delta_0\), then there exist \(C>0\) and \(n_0\geq 1\) such that
\[
\int_\Omega\left|f-\int_\Omega f\right|^p\leq C\int_\Omega\int_\Omega\frac{|f(x)-f(y)|^p}{|x-y|^p}\rho_n(|x-y|)dx\,dy \quad\forall f\in L^p(\Omega) \quad\forall n\geq n_0.\tag{1}
\]
The above estimate was suggested by some recent work of J. Bourgain, H. Brézis and P. Mironescu [in Menaldi, José Luis (ed.) et al., Optimal control and partial differential equations. In honour of Professor Alain Bensoussan’s 60th birthday. Proceedings of the conference, Paris, France, December 4, 2000. Amsterdam: IOS Press; Tokyo: Ohmsha, 439–455 (2001; Zbl 1103.46310)]. As \(n\to\infty\) in (1), we recover Poincaré’s inequality. We also extend a compactness result of Bourgain, Brézis and Mironescu.
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 26D15 | Inequalities for sums, series and integrals |
| 35J20 | Variational methods for second-order elliptic equations |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
Citations:
Zbl 1103.46310References:
| [1] | J. Bourgain, H. Brezis, personal communication; 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., Optimal Control and Partial Differential Equations (2001), IOS Press), 439-455, A volume in honour of A. Benssoussan’s 60th birthday · Zbl 1103.46310 |
| [3] | Bourgain, J.; Brezis, H.; Mironescu, P., Limiting embedding theorems for \(W^{s,p}\) when \(s\)↑1 and applications, J. Anal. Math., 87, 77-101 (2002), Dedicated to the memory of Thomas H. Wolff · Zbl 1029.46030 |
| [4] | J. Bourgain, H. Brezis, P. Mironescu, \(H^{1/2}\); J. Bourgain, H. Brezis, P. Mironescu, \(H^{1/2}\) · Zbl 1051.49030 |
| [5] | Brezis, H., How to recognize constant functions. Connections with Sobolev spaces, Uspekhi Mat. Nauk. Uspekhi Mat. Nauk, Russian Math. Surveys, 57, 693-708 (2002), Volume in honor of M. Vishik · Zbl 1072.46020 |
| [6] | Maz’ya, V.; Shaposhnikova, T., On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal.. J. Funct. Anal., J. Funct. Anal., 201, 298-300 (2003), Erratum |
| [7] | A.C. Ponce, An estimate in the spirit of Poincaré’s inequality, J. Eur. Math. Soc., in press; A.C. Ponce, An estimate in the spirit of Poincaré’s inequality, J. Eur. Math. Soc., in press |
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