Mathematical analysis
Profile decomposition and phase control for circle-valued maps in one dimension
Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1087-1092.

When 1<p<, maps f in W1/p,p((0,1);S1) have W1/p,p phases φ, but the W1/p,p-seminorm of φ is not controlled by the one of f. Lack of control is illustrated by “the kink”: f=eıφ, where the phase φ moves quickly from 0 to 2π. A similar situation occurs for maps f:S1S1, with Moebius maps playing the role of kinks. We prove that this is the only loss of control mechanism: each map f:S1S1 satisfying |f|W1/p,ppM can be written as f=eıψj=1K(Maj)±1, where Maj is a Moebius map vanishing at ajD, while the integer K=K(f) and the phase ψ are controlled by M. In particular, we have KcpM for some cp. When p=2, we obtain the sharp value of c2, which is c2=1/(4π2). As an application, we obtain the existence of minimal maps of degree one in W1/p,p(S1;S1) with p(2ε,2).

Si 1<p<, les applications f appartenant à W1/p,p((0,1);S1) ont des phases φ dans W1/p,p, mais la seminorme W1/p,p de φ n'est pas contrôlée par celle de f. L'absence de contrôle est illustrée par « le pli » : f=eıφ, où la phase φ augmente rapidement de 0 à 2π. Pour des applications f:S1S1, le même phénomène apparaît, avec les transformations de Moebius jouant le rôle des plis. Nous prouvons que cet exemple est essentiellement le seul : toute application f:S1S1 telle que |f|W1/p,ppM s'écrit f=eıψj=1K(Maj)±1, où Maj est une transformation de Moebius s'annulant en ajD, tandis que l'entier K=K(f) et la phase ψ sont contrôlés par M. En particulier, nous avons KcpM pour une constante cp. Pour p=2, nous obtenons la valeur optimale de c2, qui est c2=1/(4π2). Comme application, nous obtenons l'existence d'une application minimale de degré un dans W1/p,p(S1;S1) avec p]2ε,2[.

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DOI: 10.1016/j.crma.2015.09.030
Mironescu, Petru 1

1 Université de Lyon, Université Lyon-1, CNRS UMR 5208, Institut Camille-Jordan, 43 bd du 11-Novembre-1918, 69622 Villeurbanne cedex, France
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Mironescu, Petru. Profile decomposition and phase control for circle-valued maps in one dimension. Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1087-1092. doi : 10.1016/j.crma.2015.09.030. http://www.numdam.org/articles/10.1016/j.crma.2015.09.030/

[1] Bahouri, H.; Cohen, A.; Koch, G. A general wavelet-based profile decomposition in the critical embedding of function spaces, Confluentes Math., Volume 3 (2011) no. 3, pp. 387-411

[2] Berlyand, L.V.; Mironescu, P.; Rybalko, V.; Sandier, E. Minimax critical points in Ginzburg–Landau problems with semi-stiff boundary conditions: existence and bubbling, Commun. Partial Differ. Equ., Volume 39 (2014) no. 5, pp. 946-1005

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

[4] Bourgain, J.; Brézis, H. On the equation divY=f and application to control of phases, J. Amer. Math. Soc., Volume 16 (2003) no. 2, pp. 393-426 (electronic)

[5] Bourgain, J.; Brézis, H.; Mironescu, P. Lifting, degree, and distributional Jacobian revisited, Commun. Pure Appl. Math., Volume 58 (2005) no. 4, pp. 529-551

[6] Brézis, H. Problèmes de convergence dans certaines EDP non linéaires et applications géométriques, Goulaouic–Meyer–Schwartz Seminar, 1983–1984, Exp. No. 14, École polytechnique, Palaiseau, France, 1984 (11 p)

[7] Brézis, H.; Coron, J.-M. Convergence de solutions de H-systèmes et application aux surfaces à courbure moyenne constante, C. R. Acad. Sci. Paris, Ser. I, Volume 298 (1984) no. 16, pp. 389-392

[8] Brézis, H.; Coron, J.-M. Convergence of solutions of H-systems or how to blow bubbles, Arch. Ration. Mech. Anal., Volume 89 (1985) no. 1, pp. 21-56

[9] Brézis, H.; Lieb, E.H. A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., Volume 88 (1983) no. 3, pp. 486-490

[10] H. Brézis, P. Mironescu, Sobolev maps with values into the circle, Birkhäuser, in preparation.

[11] Brézis, H.; Nirenberg, L. Degree theory and BMO. I. Compact manifolds without boundaries, Sel. Math. New Ser., Volume 1 (1995) no. 2, pp. 197-263

[12] Gérard, P. Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., Volume 3 (1998), pp. 213-233 (electronic)

[13] Giaquinta, M.; Modica, G.; Souček, J. Cartesian currents in the calculus of variations. I. Cartesian currents, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics, vol. 37, Springer-Verlag, Berlin, 1998

[14] Giaquinta, M.; Modica, G.; Souček, J. On sequences of maps into S1 with equibounded W1/2 energies, Sel. Math. New Ser., Volume 10 (2004) no. 3, pp. 359-375

[15] Jaffard, S. Analysis of the lack of compactness in the critical Sobolev embeddings, J. Funct. Anal., Volume 161 (1999) no. 2, pp. 384-396

[16] Lions, P.-L. The concentration–compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoam., Volume 1 (1985) no. 2, pp. 45-121

[17] Mironescu, P.; Pisante, A. A variational problem with lack of compactness for H1/2(S1;S1) maps of prescribed degree, J. Funct. Anal., Volume 217 (2004) no. 2, pp. 249-279

[18] Nguyen, H.-M. Inequalities related to liftings and applications, C. R. Acad. Sci. Paris, Ser. I, Volume 346 (2008), pp. 957-962

[19] Sacks, J.; Uhlenbeck, K. The existence of minimal immersions of 2-spheres, Ann. Math. (2), Volume 113 (1981) no. 1, pp. 1-24

[20] Struwe, M. A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., Volume 187 (1984) no. 4, pp. 511-517

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