We sharpen an estimate of Bourgain, Brezis, and Nguyen for the topological
degree of continuous maps from a sphere $\mathbb{S}^d$ into itself in the case
$d \ge 2$. This provides the answer for $d \ge 2$ to a question raised by
Brezis. The problem is still open for $d=1$

Nguyen, Hoai-Minh

English

arXiv

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 355 (2017) 1046–1049Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comMathematical analysis/Differential topologyA refined estimate for the topological degreeUne estimée raffinée du degré topologiqueHoai-Minh NguyenÉcole polytechnique fédérale de Lausanne, EPFL, SB MATHAA CAMA, Station 8, CH-1015 Lausanne, Switzerlanda r t i c l e i n f o a b s t r a c tArticle history:Received 8 October 2017Accepted 12 October 2017Available online 16 October 2017Presented by Haïm BrézisWe sharpen an estimate of [4] for the topological degree of continuous maps from a sphere Sd into itself in the case d ≥ 2. This provides the answer for d ≥ 2 to a question raised by Brezis. The problem is still open for d = 1.© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous affinons une estimée du degré topologique pour des applications continues d’une sphère Sd dans elle-même dans le cas d ≥ 2. Cela fournit la réponse pour d ≥ 2 à une question posée par Brezis. Le problème est encore ouvert pour d = 1.© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionMotivated by the theory of Ginzburg–Landau equations (see, e.g., [1]), Bourgain, Brezis and the author established in [4]the following theorem.Theorem 1. Let d ≥ 1. For every 0 < δ < √2, there exists a positive constant C(δ) such that, for all g ∈ C(Sd, Sd),|deg g| ≤ C(δ)ˆSdˆSd|g(x)−g(y)|>δ1|x − y|2d dx dy. (1)Here and in what follows, for x ∈ Rd+1, |x| denotes its Euclidean norm in Rd+1.The constant C(δ) depends also on d, but for simplicity of notation, we omit d. Estimate (1) was initially suggested by Bourgain, Brezis, and Mironescu in [2]. It was proved in [3] in the case where d = 1 and δ is sufficiently small. In [9], the E-mail address: hoai-minh.nguyen@epfl.ch.https://doi.org/10.1016/j.crma.2017.10.0071631-073X/© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.H.-M. Nguyen / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 1046–1049 1047author improved (1) by establishing that (1) holds for 0 < δ < d =√2 + 2d+1 with a constant C(δ) independent of δ. It was also shown there that (1) does not hold for δ ≥ d .This note is concerned with the behavior of C(δ) as δ → 0. Brezis [7] (see also [6, Open problem 3]) conjectured that (1)holds withC(δ) = Cδd, (2)for some positive constant C depending only on d. This conjecture is somehow motivated by the fact that (1)–(2) holds “in the limit” as δ → 0. More precisely, it is known that (see [8, Theorem 2])limδ→0ˆSdˆSd|g(x)−g(y)|>δδd|x − y|2d dx dy = KdˆSd|∇g(x)|d dx for g ∈ C1(Sd)for some positive constant Kd depending only on d, and thatdeg g = 1|Sd|ˆSdJac(g) for g ∈ C1(Sd,Sd),by Kronecker’s formula.In this note, we confirm Brezis’ conjecture for d ≥ 2. The conjecture is still open for d = 1. Here is the result of the note.Theorem 2. Let d ≥ 2. There exists a positive constant C = C(d), depending only on d, such that, for all g ∈ C(Sd, Sd),|deg g| ≤ CˆSdˆSd|g(x)−g(y)|>δδd|x − y|2d dx dy for 0 < δ < 1. (3)2. Proof of Theorem 2The proof of Theorem 2 is in the spirit of the approach in [4,9]. One of the new ingredients of the proof is the following result [10, Theorem 1], which has its roots in [5]:Lemma 1. Let d ≥ 1, p ≥ 1, let B be an open ball in Rd, and let f be a real bounded measurable function defined in B. We have, for all δ > 0,1|B|2ˆBˆB| f (x) − f (y)|p dx dy ≤ C p,d⎛⎜⎜⎜⎝|B|pd −1ˆBˆB| f (x)− f (y)|>δδp|x − y|d+p dx dy + δp⎞⎟⎟⎟⎠ , (4)for some positive constant C p,d depending only on p and d.In Lemma 1, |B| denotes the Lebesgue measure of B .We are ready to presentProof of Theorem 2. We follow the strategy in [4,9]. We first assume in addition that g ∈ C1(Sd, Sd). Let B be the open unit ball in Rd+1 and let u : B → B be the average extension of g , i.e.u(X) = −ˆB(x,r)g(s)ds for X ∈ B, (5)where x = X/|X |, r = 2(1 − |X |), and B(x, r) := {y ∈ Sd; |y − x| ≤ r}. In this proof, −́D g(s) ds denotes the quantity 1|D|´D g(s) ds for a measurable subset D of Sd with positive (d-dimensional Hausdorff) measure. Fix α = 1/2 and for every x ∈ Sd , let ρ(x) be the length of the largest radial interval coming from x on which |u| > α (possibly ρ(x) = 1). In particular, if ρ(x) < 1, then∣∣∣∣∣∣∣−ˆB(x,2ρ(x))g(s)ds∣∣∣∣∣∣∣= 1/2. (6)1048 H.-M. Nguyen / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 1046–1049By [4, (7)], we have|deg g| ≤ CˆSdρ(x)<11ρ(x)ddx. (7)Here and in what follows, C denotes a positive constant, which is independent of x, ξ , η, g , and δ, and can change from one place to another.We now implement ideas involving Lemma 1 applied with p = 1. We have, by (6),−ˆB(x,2ρ(x))−ˆB(x,2ρ(x))|g(ξ) − g(η)|dξ dη ≥ −ˆB(x,2ρ(x))∣∣∣∣∣∣∣g(ξ) − −ˆB(x,2ρ(x))g(η)dη∣∣∣∣∣∣∣dξ ≥ C .This yields, for some 1 ≤ j0 ≤ d + 1,−ˆB(x,2ρ(x))−ˆB(x,2ρ(x))|g j0(ξ) − g j0(η)|dξ dη ≥ C,where g j denotes the j-th component of g . It follows from (4) that, for some δ0 > 0 (δ0 depends only on d) and for 0 < δ < δ0,ρ(x)1−dˆB(x,2ρ(x))ˆB(x,2ρ(x))|g j0 (ξ)−g j0 (η)|>δδ|ξ − η|d+1 dξ dη ≥ C,which impliesd+1∑j=1ρ(x)1−dˆB(x,2ρ(x))ˆB(x,2ρ(x))|g j(ξ)−g j(η)|>δδ|ξ − η|d+1 dξ dη ≥ C . (8)Sinceρ(x)1−dˆB(x,2ρ(x))ˆB(x,2ρ(x))|ξ−η|>C1ρ(x)δδ|ξ − η|d+1 dξ dη <C2(d + 1) ,if C1 > 0 is large enough (the largeness of C1 depends only on C and d), it follows from (8) thatd+1∑j=1ρ(x)1−dˆB(x,2ρ(x))ˆB(x,2ρ(x))|g j(ξ)−g j(η)|>δ|ξ−η|≤Cρ(x)δδ|ξ − η|d+1 dξ dη ≥ C . (9)We derive from (7) and (9) that, for 0 < δ < δ0,|deg g| ≤ CˆSdρ(x)<11ρ(x)2d−1dxd+1∑j=1ˆB(x,2ρ(x))ˆB(x,2ρ(x))|g j(ξ)−g j(η)|>δ|ξ−η|≤Cρ(x)δδ|ξ − η|d+1 dξ dη.This implies, by Fubini’s theorem, that, for 0 < δ < δ0,|deg g| ≤ Cd+1∑j=1ˆSdˆSd|g j(ξ)−g j(η)|>δδ|ξ − η|d+1 dξ dηˆρ(x)≥C |ξ−η|/δ2ρ(x)>|x−ξ |1ρ(x)2d−1dx. (10)We haveH.-M. Nguyen / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 1046–1049 1049ˆ2ρ(x)>|x−ξ |ρ(x)≥C |ξ−η|/δ1ρ(x)2d−1dx ≤ˆ2ρ(x)>|x−ξ ||x−ξ |>C |ξ−η|/δ1ρ(x)2d−1dx +ˆρ(x)≥C |ξ−η|/δ|x−ξ |≤C |ξ−η|/δ1ρ(x)2d−1dx≤ˆ|x−ξ |>C |ξ−η|/δC|x − ξ |2d−1 dx +ˆ|x−ξ |≤C |ξ−η|/δCδ2d−1|ξ − η|2d−1 dx.Finally, we use the assumption that d ≥ 2. Since d > 1, it follows thatˆρ(x)>|x−ξ |ρ(x)≥C |ξ−η|/δ1ρ(x)2d−1dx ≤ Cδd−1|ξ − η|d−1 . (11)Combining (10) and (11) yields, for 0 < δ < δ0,|deg g| ≤ Cd+1∑j=1ˆSdˆSd|g j(ξ)−g j(η)|>δδd|ξ − η|2d dξ dη. (12)Assertion (3) is now a direct consequence of (12) for δ < δ0 and (1) for δ0 ≤ δ < 1.The proof in the case g ∈ C(Sd, Sd) can be derived from the case g ∈ C1(Sd, Sd) via a standard approximation argument. The details are omitted. AcknowledgementsThe author warmly thanks Haïm Brezis for communicating [7], and Haïm Brezis and Itai Shafrir for interesting discus-sions.References[1] F. Bethuel, H. Brezis, F. Helein, Ginzburg–Landau Vortices, Prog. Nonlinear Differ. Equ. Appl., vol. 13, Birkhäuser, Boston, 1994.[2] J. Bourgain, H. Brezis, P. Mironescu, Lifting, degree, and distributional Jacobian revisited, Commun. Pure Appl. Math. 58 (2005) 529–551.[3] J. Bourgain, H. Brezis, P. Mironescu, Complements to the paper “Lifting, Degree, and Distributional Jacobian Revisited”, https://hal.archives-ouvertes.fr/hal-00747668/document.[4] J. Bourgain, H. Brezis, H-M. Nguyen, A new estimate for the topological degree, C. R. Acad. Sci. Paris, Ser. I 340 (2005) 787–791.[5] J. Bourgain, H-M. Nguyen, A new characterization of Sobolev spaces, C. R. Acad. Sci. Paris, Ser. I 343 (2006) 75–80.[6] H. Brezis, New Questions Related to the Topological Degree. The Unity of Mathematics, Prog. Math., vol. 244, Birkhäuser, 2006, pp. 137–154.[7] H. Brezis, Private communication, 2006.[8] H-M. Nguyen, Some new characterizations of Sobolev spaces, J. Funct. Anal. 237 (2006) 689–720.[9] H-M. Nguyen, Optimal constant in a new estimate for the degree, J. Anal. Math. 101 (2007) 367–395.[10] H-M. Nguyen, Some inequalities related to Sobolev norms, Calc. Var. Partial Differ. Equ. 41 (2011) 483–509.

A refined estimate for the topological degree

Hoai-Minh Nguyen

Comptes Rendus Mathématique

We sharpen an estimate of [4] for the topological degree of continuous maps from a sphere Sdinto itself in the case d >= 2. This provides the answer for d >= 2 to a question raised by Brezis. The problem is still open for d = 1. (C) 2017 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.CAM

Nguyên, Hoài-Minh

Infoscience - École polytechnique fédérale de Lausanne

A refined estimate for the topological degreeHoai-Minh Nguyen∗October 8, 2017AbstractWe sharpen an estimate of [4] for the topological degree of continuous maps from asphere Sd into itself in the case d ≥ 2. This provides the answer for d ≥ 2 to a questionraised by Brezis. The problem is still open for d = 1.AMS classification: 47H11, 55C25, 58C35.Keywords: topological degree, fractional Sobolev spaces.1 IntroductionMotivated by the theory of Ginzburg Landau equations (see, e.g., [1]), Bourgain, Brezis,and the author established in [4]:Theorem 1. Let d ≥ 1. For every 0 < δ <√2, there exists a positive constant C(δ) such that,for all g ∈ C(Sd,Sd), ∣deg g∣ ≤ C(δ) ∫Sd∫Sd∣g(x)−g(y)∣>δ1∣x − y∣2d dxdy. (1.1)Here and in what follows, for x ∈ Rd+1, ∣x∣ denotes its Euclidean norm in Rd+1.The constant C(δ) depends also on d but for simplicity of notation we omit d. Esti-mate (1.1) was initially suggested by Bourgain, Brezis, and Mironescu in [2]. It was provedin [3] in the case where d = 1 and δ is sufficiently small. In [9], the author improved (1.1) byestablishing that (1.1) holds for 0 < δ < `d = √2 + 2d+1 with a constant C(δ) independent ofδ. It was also shown there that (1.1) does not hold for δ ≥ `d.This note is concerned with the behavior of C(δ) as δ → 0. Brezis [7] (see also [6, Openproblem 3]) conjectured that (1.1) holds withC(δ) = Cδd, (1.2)∗EPFL SB MATHAA CAMA, Station 8, CH-1015 Lausanne, Switzerland, hoai-minh.nguyen@epfl.ch1for some positive constant C depending only on d. This conjecture is somehow motivated bythe fact that (1.1)-(1.2) holds “in the limit” as δ → 0. More precisely, it is known that (see[8, Theorem 2])limδ→0 ∫Sd ∫Sd∣g(x)−g(y)∣>δδd∣x − y∣2d dxdy =Kd∫Sd ∣∇g(x)∣d dx for g ∈ C1(Sd)for some positive constant Kd depending only on d and thatdeg g = 1∣Sd∣ ∫Sd Jac(g) for g ∈ C1(Sd,Sd),by Kronecker’s formula.In this note, we confirm Brezis’ conjecture for d ≥ 2. The conjecture is still open for d = 1.Here is the result of the note.Theorem 2. Let d ≥ 2. There exists a positive constant C = C(d), depending only on d, suchthat, for all g ∈ C(Sd,Sd),∣deg g∣ ≤ C ∫Sd∫Sd∣g(x)−g(y)∣>δδd∣x − y∣2d dxdy for 0 < δ < 1. (1.3)2 Proof of Theorem 2The proof of Theorem 2 is in the spirit of the approach in [4, 9]. One of the new ingredientsof the proof is the following result [10, Theorem 1], which has its roots in [5]:Lemma 1. Let d ≥ 1, p ≥ 1, let B be an open ball in Rd, and let f be a real bounded measurablefunction defined in B. We have, for all δ > 0,1∣B∣2 ∫B ∫B ∣f(x) − f(y)∣p dxdy ≤ Cp,d ⎛⎜⎜⎝∣B∣pd−1 ∫B∫B∣f(x)−f(y)∣>δδp∣x − y∣d+p dxdy + δp⎞⎟⎟⎠ , (2.1)for some positive constant Cp,d depending only on p and d.In Lemma 1, ∣B∣ denotes the Lebesgue measure of B.We are ready to presentProof of Theorem 2. We follow the strategy in [4, 9]. We first assume in addition thatg ∈ C1(Sd,Sd). Let B be the open unit ball in Rd+1 and let u ∶ B → B be the average extensionof g, i.e.,u(X) = ⨏B(x,r) g(s)ds for X ∈ B, (2.2)where x =X/∣X ∣, r = 2(1− ∣X ∣), and B(x, r) ∶= {y ∈ Sd; ∣y−x∣ ≤ r}. In this proof, ⨏D g(s)ds de-notes the equantity 1∣D∣ ∫D g(s)ds for a measurable subset D of Sd with positive (d-dimensional2Hausdorff) measure. Fix α = 1/2 and for every x ∈ Sd, let ρ(x) be the length of the largestradial interval coming from x on which ∣u∣ > α (possibly ρ(x) = 1). In particular, if ρ(x) < 1,then ∣⨏B(x,2ρ(x)) g(s)ds∣ = 1/2. (2.3)By [4, (7)], we have ∣deg g∣ ≤ C ∫Sdρ(x)<11ρ(x)d dx. (2.4)Here and in what follows, C denotes a positive constant which is independent of x, ξ , η, g,and δ, and can change from one place to another.We now implement ideas involving Lemma 1 applied with p = 1. We have, by (2.3),⨏B(x,2ρ(x))⨏B(x,2ρ(x)) ∣g(ξ) − g(η)∣dξ dη ≥ ⨏B(x,2ρ(x)) ∣g(ξ) − ⨏B(x,2ρ(x)) g(η)dη∣ dξ ≥ C.This yields, for some 1 ≤ j0 ≤ d + 1,⨏B(x,2ρ(x))⨏B(x,2ρ(x)) ∣gj0(ξ) − gj0(η)∣dξ dη ≥ C,where gj denotes the j-th component of g. It follows from (2.1) that, for some δ0 > 0 (δ0depends only on d) and for 0 < δ < δ0,ρ(x)1−d∫B(x,2ρ(x))∫B(x,2ρ(x))∣gj0(ξ)−gj0(η)∣>δδ∣ξ − η∣d+1 dξ dη ≥ C,which impliesd+1∑j=1 ρ(x)1−d∫B(x,2ρ(x))∫B(x,2ρ(x))∣gj(ξ)−gj(η)∣>δδ∣ξ − η∣d+1 dξ dη ≥ C. (2.5)Sinceρ(x)1−d∫B(x,2ρ(x))∫B(x,2ρ(x))∣ξ−η∣>C1ρ(x)δδ∣ξ − η∣d+1 dξ dη < C2(d + 1) ,if C1 > 0 is large enough (the largeness of C1 depends only on C and d), it follows from (2.5)thatd+1∑j=1 ρ(x)1−d∫B(x,2ρ(x))∫B(x,2ρ(x))∣gj(ξ)−gj(η)∣>δ∣ξ−η∣≤Cρ(x)δδ∣ξ − η∣d+1 dξ dη ≥ C. (2.6)We derive from (2.4) and (2.6) that, for 0 < δ < δ0,∣deg g∣ ≤ C ∫Sdρ(x)<11ρ(x)2d−1 dx d+1∑j=1∫B(x,2ρ(x))∫B(x,2ρ(x))∣gj(ξ)−gj(η)∣>δ∣ξ−η∣≤Cρ(x)δδ∣ξ − η∣d+1 dξ dη.3This implies, by Fubini’s theorem, that, for 0 < δ < δ0,∣deg g∣ ≤ C d+1∑j=1 ∫Sd ∫Sd∣gj(ξ)−gj(η)∣>δδ∣ξ − η∣d+1 dξ dη ∫ρ(x)≥C∣ξ−η∣/δ2ρ(x)>∣x−ξ∣1ρ(x)2d−1 dx. (2.7)We have∫2ρ(x)>∣x−ξ∣ρ(x)≥C∣ξ−η∣/δ1ρ(x)2d−1 dx ≤ ∫2ρ(x)>∣x−ξ∣∣x−ξ∣>C∣ξ−η∣/δ1ρ(x)2d−1 dx + ∫ρ(x)≥C∣ξ−η∣/δ∣x−ξ∣≤C∣ξ−η∣/δ1ρ(x)2d−1 dx≤ ∫∣x−ξ∣>C∣ξ−η∣/δC∣x − ξ∣2d−1 dx + ∫∣x−ξ∣≤C∣ξ−η∣/δCδ2d−1∣ξ − η∣2d−1 dx.Finally, we use the assumption that d ≥ 2. Since d > 1, it follows that∫ρ(x)>∣x−ξ∣ρ(x)≥C∣ξ−η∣/δ1ρ(x)2d−1 dx ≤ Cδd−1∣ξ − η∣d−1 . (2.8)Combining (2.7) and (2.8) yields, for 0 < δ < δ0,∣deg g∣ ≤ C d+1∑j=1 ∫Sd ∫Sd∣gj(ξ)−gj(η)∣>δδd∣ξ − η∣2d dξ dη. (2.9)Assertion (1.3) is now a direct consequence of (2.9) for δ < δ0 and (1.1) for δ0 ≤ δ < 1.The proof in the case g ∈ C(Sd,Sd) can be derived from the case g ∈ C1(Sd,Sd) via astandard approximation argument. The details are omitted.Acknowledgement: The author warmly thanks Haim Brezis for communicating [7] andHaim Brezis and Itai Shafrir for interesting discussions.References[1] F. Bethuel, H. Brezis, F. Helein, Ginzburg-Landau vortices. Progress in Nonlinear Dif-ferential Equations and their Applications, 13, Birkha¨user Boston, 1994.[2] J. Bourgain, H. Brezis, P. Mironescu, Lifting, Degree, and Distributional Jacobian Re-visited, Comm. Pure Appl. Math., 58 (2005), 529-551.[3] J. Bourgain, H. Brezis, P. Mironescu, Complements to the paper ” Lifting, De-gree, and Distributional Jacobian Revisited”, https://hal.archives-ouvertes.fr/hal-00747668/document.4[4] J. Bourgain, H. Brezis, H-M. Nguyen, A new estimate for the topological degree, C. R.Math. Acad. Sci. Paris 340 (2005), 787–791.[5] J. Bourgain and H-M. Nguyen, A new characterization of Sobolev spaces, C. R. Acad.Sci. Paris 343 (2006), 75-80.[6] H. Brezis, New questions related to the topological degree. The unity of mathematics,137–154, Progr. Math. 244, Birkhauser, 2006.[7] H. Brezis, Private communication, 2006.[8] H-M. Nguyen, Some new characterizations of Sobolev spaces, J. Funct. Anal. 237 (2006),689–720.[9] H-M. Nguyen, Optimal constant in a new estimate for the degree, J. Anal. Math. 101(2007), 367–395.[10] H-M. Nguyen, Some inequalities related to Sobolev norms, Calc. Var. Partial DifferentialEquations 41 (2011), 483–509.5

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A refined estimate for the topological degree

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