Comptes Rendus
Mathematical Analysis
Decomposition of S1-valued maps in Sobolev spaces
Comptes Rendus. Mathématique, Volume 348 (2010) no. 13-14, pp. 743-746.

Let n2, s>0, p1 be such that 1sp<2. We prove that for each map uWs,p(Sn;S1) one can find φWs,p(Sn;R) and vWsp,1(Sn;S1) such that u=veıφ. This yields a decomposition of u into a part that has a lifting in Ws,p, eıφ, and a map “smoother” than u but without lifting, namely v. Our result generalizes a previous one of Bourgain and Brezis (which corresponds to the case s=1/2, p=2). As a consequence, we find an intuitive proof for the existence of the distributional Jacobian Ju of maps uWs,p(Sn;S1) (originally due to Bourgain, Brezis and the author). By completing a result of Bousquet, we characterize the distributions of the form Ju.

Soient n2, s>0, p1 tels que 1sp<2. Nous montrons que, pour chaque uWs,p(Sn;S1), il existe φWs,p(Sn;R) et vWsp,1(Sn;S1) tels que u=veıφ. Ceci donne une décomposition de u comme produit d'un facteur qui se relève dans Ws,p, eıφ, et d'un facteur « plus régulier » que u mais qui ne se relève pas, à savoir v. Notre décomposition généralise un résultat antérieur de Bourgain et Brezis (qui ont traité le cas s=1/2, p=2). Une conséquence de notre résultat est une preuve intuitive de l'existence du jacobien au sens des distributions Ju pour les applications uWs,p(Sn;S1) (résultat dû, avec un argument différent, à Bourgain, Brezis et l'auteur). En complétant un résultat de Bousquet, nous caractérisons les distributions de la forme Ju.

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DOI: 10.1016/j.crma.2010.06.020

Petru Mironescu 1

1 Université de Lyon, CNRS, Université Lyon 1, Institut Camille-Jordan, 43, boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France
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Petru Mironescu. Decomposition of $ {\mathbb{S}}^{1}$-valued maps in Sobolev spaces. Comptes Rendus. Mathématique, Volume 348 (2010) no. 13-14, pp. 743-746. doi : 10.1016/j.crma.2010.06.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.06.020/

[1] G. Alberti; S. Baldo; G. Orlandi Functions with prescribed singularities, J. Eur. Math. Soc., Volume 5 (2003) no. 3, pp. 275-311

[2] J.M. Ball Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., Volume 63 (1977), pp. 337-403

[3] F. Bethuel; X. Zheng Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal., Volume 80 (1988), pp. 60-75

[4] J. Bourgain; H. Brezis On the equation divY=f and application to control of phases, J. Amer. Math. Soc., Volume 16 (2003), pp. 393-426

[5] J. Bourgain; H. Brezis; P. Mironescu Lifting in Sobolev spaces, J. Anal. Math., Volume 80 (2000), pp. 37-86

[6] J. Bourgain; H. Brezis; P. Mironescu H1/2 maps with values into the circle: minimal connections, lifting, and the Ginzburg–Landau equation, Publ. Math. Inst. Hautes Études Sci., Volume 99 (2004), pp. 1-115

[7] P. Bousquet Topological singularities in Ws,p(SN,S1), J. Anal. Math., Volume 102 (2007), pp. 311-346

[8] P. Bousquet, P. Mironescu, in preparation

[9] H. Brezis; J.-M. Coron; E. Lieb Harmonic maps with defects, Comm. Math. Phys., Volume 107 (1986) no. 4, pp. 649-705

[10] H. Brezis; P. Mironescu Gagliardo–Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ., Volume 1 (2001), pp. 387-404

[11] H. Brezis; P. Mironescu; A. Ponce W1,1-maps with values into S1 (S. Chanillo; P.D. Cordaro; N. Hanges; J. Hounie; A. Meziani, eds.), Geometric Analysis of PDE and Several Complex Variables, Contemporary Mathematics, vol. 368, Amer. Math. Soc., Providence, RI, 2005, pp. 69-100

[12] R.L. Jerrard; H.M. Soner Functions of bounded higher variation, Indiana Univ. Math. J., Volume 51 (2002), pp. 645-677

[13] V. Maz'ya; T. Shaposhnikova An elementary proof of the Brezis and Mironescu theorem on the composition operator in fractional Sobolev spaces, J. Evol. Equ., Volume 2 (2002) no. 1, pp. 113-125

[14] P. Mironescu Lifting default for S1-valued maps, C. R. Acad. Sci. Paris, Ser. I, Volume 346 (2008) no. 19–20, pp. 1039-1044

[15] P. Mironescu S1-valued Sobolev maps http://math.univ-lyon1.fr/~mironescu/7.pdf

[16] P. Mironescu, Sobolev spaces of circle-valued maps, in preparation

[17] C.B. Morrey Multiple Integrals in the Calculus of Variations, Die Grundlehren der mathematischen Wissenschaften, vol. 130, Springer-Verlag New York, Inc., New York, 1966

[18] H.-M. Nguyen Inequalities related to liftings and applications, C. R. Acad. Sci. Paris, Ser. I, Volume 346 (2008) no. 17–18, pp. 957-962

[19] Y.G. Reshetnyak The weak convergence of completely additive vector-valued set functions, Sibirsk. Mat. Zh., Volume 9 (1968), pp. 1386-1394

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