Une preuve simple d’une inégalite ́ de Bourgain, Brezis et

A simple proof of an inequality of Bourgain, Brezis and Mironescu Une preuve simple d’une inégalité de Bourgain, Brezis et Mironescu Jean Van Schaftingen a a Département de Mathématique, Université catholique de Louvain, 2, chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium Abstract A simpler proof of a recent inequality of Bourgain, Brezis and Mironescu is given. To cite this article: J. Van Schaftingen, C. R. Acad. Sci. Paris, Ser. I 336 (2003). Résumé Nous donnons une preuve plus simple d’une inégalité récente de Bourgain, Brezis et Mironescu. Pour citer cet article : J. Van Schaftingen, C. R. Acad. Sci. Paris, Ser. I 336 (2003). Version française abrégée Bourgain, Brezis et Mironescu ont établi dans [1] l’inégalité suivante. Proposition 0.1 Soit Γ une courbe fermée, orientée et rectifiable de R3 , et soit ~t le vecteur unité tangent à Γ. Si ~k ∈ W 1,3 (R3 ; R3 ), alors Z ~k · ~t ≤ C kkk 1,3 |Γ| . W Γ La preuve de la proposition 0.1 dans [1] est assez complexe. Nous en donnons une preuve élémentaire qui se généralise à des surfaces de dimension quelconque dans un espace de dimension quelconque. Email address: [email protected] (Jean Van Schaftingen). Preprint submitted to Elsevier Science 15 octobre 2003 Proposition 0.1 Soit Γ une courbe fermée, orientée et lipschitzienne dans RN , N ≥ 2 ; soit u ∈ 1,1 Wloc (RN ). Si ∇u ∈ LN (RN ), alors Z u dγ ≤ CN k∇ukLN (RN ) |Γ| , (1) Γ où |Γ| désigne la longueur de la courbe Γ. La preuve de 0.1 commence par établir, avec la stratégie de [1], la formule (3). Ensuite, alors que Bourgain, Brezis et Mironescu utilisent une décomposition de Littlewood-Paley, nous utilisons les inégalités de Morrey et de Hölder pour conclure. La preuve se généralise sans difficulté à des surfaces k-dimensionnelles (proposition 3.2). 1. Introduction Bourgain, Brezis and Mironescu proved in [1] the following inequality. Proposition 1.1 Let Γ be a closed, oriented, rectifiable curve of R3 , and denote by ~t the unit tangent vector along Γ; let ~k ∈ W 1,3 (R3 ; R3 ). Then Z ~k · ~t ≤ C kkk 1,3 |Γ| . W Γ The proof of proposition 1.1 in [1] is technically involved. We provide an elementary proof and a generalization to k dimensional surfaces and N -dimensional space. For simplicity, we begin with the case of a curve in RN . Proposition 1.2 Let Γ be an oriented, compact and closed Lipschitz curve of RN , N ≥ 2; let u ∈ 1,1 Wloc (RN ). If ∇u ∈ LN (RN ), then Z u dγ ≤ CN k∇ukLN (RN ) |Γ| , (2) Γ where |Γ| denotes the length of curve Γ. Remark 1 When N = 1, the left-hand side of (2) is 0; when N = 2 and γ is a Jordan curve, proposition 1.2 is a simple consequence of Green theorem and the isoperimetric inequality; and when N = 3 it is equivalent to proposition 1.1. 2. Proof of proposition 1.2 PROOF. Without loss of generality, the curve Γ is connected and is the image of S 1 by a Lipschitz map γ. We assume first that u and γ : S 1 → RN are of class C 1 . We start with the same strategy as [1], bounding Z e · u(γ(x))γ̇(x) dx, S1 for an arbitrary unit-norm vector e ∈ RN . 2 Let  Γt = x ∈ S 1 | e · γ(x) = t . Since e · γ(s) is of class C 1 , Sard’s lemma implies that for almost every t ∈ R, Γt is finite and e · γ̇(s) 6= 0 if γ(s) ∈ Γt . We have then Z Z Z X e · u(γ(x))γ̇(x) dx = u(γ(x)) e · γ̇(x) dx = σ(x)u(x) dt , (3) S1 R x∈Γt S1  where σ(x) = sign (e · γ̇(x)). Since Γ is closed, for almost every t ∈ R we can write Γt = P1 , . . . , Pr(t) ∪  Pr N1 , . . . , Nr(t) so that σ(Pi ) = 1, σ(Ni ) = −1 and i=1 |γ(Pi ) − γ(Ni )| is minimal (in particular, it is bounded by |Γ|). P In order to estimate x∈Γt σ(x)u(x), we proceed differently from [1]. They used a Littlewood-Paley decomposition. Instead, we apply Morrey’s inequality (see e.g. [2, theorem IX.12]) in RN −1 for ut = u|{y∈RN | e·y=t} before applying the discrete Hölder inequality to the sum X σ(x)u(x) ≤ CN k∇ut kN r(t) X 1 1 |γ(Pi ) − γ(Ni )| N ≤ CN k∇ut kp |Γ| N r(t) N −1 N . i=1 x∈Γt We are now ready to estimate the integral of (3): Z Z X 1 N −1 σ(x)u(x) dt ≤ CN |Γ| N k∇ut kN r(t) N dt R x∈Γt R 1 ≤ CN |Γ| N  NN−1   N1  Z Z ′  k∇ut kN dt  r(t) dt ≤ Cp,N |Γ| k∇ukN N R R since 2 R r(t) dt = S 1 |e · γ ′ (x)| dx ≤ S 1 |γ ′ (x)| dx = |Γ|. The result is extended to general Γ and u by standard smoothing arguments. R R R 3. Generalization to surfaces Proposition 1.2 generalizes straightforwardly to k-dimensional surfaces defined as follows. Definition 3.1 A pair Γ = (M, γ) is a C r k-dimensional Lipschitz surface of RN if (i) M is a compact oriented k-dimensional C r manifold without boundary, (ii) γ : M → RN is a Lipschitz function. When Γ = (M, γ) is a k-dimensional Lipschitz surface of RN , it R is possible, since M is oriented, to define the integral of Borel function u : RN → R asRthe k-vector Γ u dγ(x), where dγ(x)[a1 , . . . , ak ] = γ ′ (x)a1 ∧ . . . ∧ γ ′ (x)ak , and the mass of Γ as |Γ| = M |dγ|, where |·| denotes the euclidean norm of a k-vector. Proposition 3.2 Let Γ be a C k k-dimensional surface. Then Z u dγ ≤ CN k∇ukN |Γ| , (4) M where the norm on the left is the comass-norm (see [3]). 3 PROOF. Since the proof is similar to the proof of proposition 1.2, we only give a sketch. For an arbitrary simple unit covector e = e1 ∧ · · · ∧ ek , we write Z X Z σ(x)u(x) dy, e · u dγ = Γ Rk x∈Γy where Γy = {x ∈ M | ei · γ(x) = yi , 1 ≤ i ≤ k} and σ(x) = sign (e1 γ ′ (x) ∧ . . . ∧ ek γ ′ (x)). This formula is valid by Sard’s lemma because M is a C k manifold. Then, using Morrey’s and Hölder’s inequalities, with the notations of the proof of proposition 1.2,  N1−1    NN−1 Z r(y) Z X Z Z X  |γ(Pi ) − γ(Ni )| dy ′′  σ(x)u(x) dy ≤ k∇uy1 k  dy1 , r(y) dy ′′  N Rk x∈Γy R Rk−1 i=1 where y ′′ R= (y2 , . . . , yk ) and one concludes using ity and 2 Rk r(y) dy ≤ |Γ|. R Rk−1 Rk−1 Pr(y) i=1 |γ(Pi ) − γ(Ni )| dy ′′ ≤ |Γ|, Hölder’s inequal- Remark 2 Proposition 3.2 can also be proved by induction on k. The case k = 1 is proposition 1.2 and for k > 1, Γ is cut into slices of dimension k − 1, for which the estimate of proposition 3.2 holds. The integration of this estimate, with Hölder’s inequality gives the conclusion. Remark 3 The inequality (4) is the limit case of Z N u dγ ≤ Cp,N δ(Γ)1− p |Γ| k∇ukp , Γ where p > N , ∇u ∈ Lp (RN ) and δ(Γ) denotes the diameter of Γ. It is a simple consequence of Morrey’s inequality. Acknowledgements The author wishes to thank Haı̈m Brezis for proposing the problem and for comments. He also acknowledges a Research Fellow grant of the Fonds National de la Recherche Scientifique. References [1] J. Bourgain, H. Brezis, and P. Mironescu, H 1/2 maps with value into the circle; minimal connections, lifting, and the Ginzburg-Landau equation, in press. [2] H. Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maı̂trise, Masson, Paris, 1983. [3] H. Federer, Geometric measure theory, Springer-Verlag, New York, 1969. 4