Inequalities related to liftings and applications

Hoai-Minh Nguyen

doi:10.1016/j.crma.2008.07.026

Partial Differential Equations

Inequalities related to liftings and applications

Presented by: Haïm Brezis

Hoai-Minh Nguyen ¹

¹ Rutgers University, Department of Mathematics, Hill Center, Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA

Comptes Rendus. Mathématique, Volume 346 (2008) no. 17-18, pp. 957-962.

Abstract
Résumé

We present two inequalities for liftings of smooth maps from the torus $T^{d}$ into the unit circle $S^{1}$ . Using these inequalities, we answer a question of J. Bourgain, H. Brezis, and P. Mironescu in [J. Bourgain, H. Brezis, P. Mironescu, Lifting, degree, and distributional Jacobian revisited, Comm. Pure Appl. Math. 58 (2005) 529–551] and establish an estimate of liftings in the spirit of R.R. Coifman and Y. Meyer [R.R. Coifman, Y. Meyer, Une généralisation du théorème de Calderon sur l'intégrale de Cauchy, in: Fourier Analysis, in: Proc. Sem., El Escorial, vol. 1, Asoc. Mat. Espa nola, Madrid, 1980, pp. 87–116].

Nous présentons deux inégalités pour des relèvements des applications régulières du tore $T^{d}$ dans le cercle unité $S^{1}$ . Ces inégalités nous permettent de répondre à une question de J. Bourgain, H. Brezis, et P. Mironescu dans [J. Bourgain, H. Brezis, P. Mironescu, Lifting, degree, and distributional Jacobian revisited, Comm. Pure Appl. Math. 58 (2005) 529–551] et d'établir une estimation des relèvements dans l'esprit de R.R. Coifman et Y. Meyer [R.R. Coifman, Y. Meyer, Une généralisation du théorème de Calderon sur l'intégrale de Cauchy, in : Fourier Analysis, in : Proc. Sem., El Escorial, vol. 1, Asoc. Mat. Espa nola, Madrid, 1980, pp. 87–116].

Accepted: 2008-07-30
Published online: 2008-08-29

DOI: 10.1016/j.crma.2008.07.026

Author's affiliations:

Hoai-Minh Nguyen ¹

¹ Rutgers University, Department of Mathematics, Hill Center, Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA

@article{CRMATH_2008__346_17-18_957_0,
     author = {Hoai-Minh Nguyen},
     title = {Inequalities related to liftings and applications},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {957--962},
     publisher = {Elsevier},
     volume = {346},
     number = {17-18},
     year = {2008},
     doi = {10.1016/j.crma.2008.07.026},
     language = {en},
}

TY  - JOUR
AU  - Hoai-Minh Nguyen
TI  - Inequalities related to liftings and applications
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 957
EP  - 962
VL  - 346
IS  - 17-18
PB  - Elsevier
DO  - 10.1016/j.crma.2008.07.026
LA  - en
ID  - CRMATH_2008__346_17-18_957_0
ER  -

%0 Journal Article
%A Hoai-Minh Nguyen
%T Inequalities related to liftings and applications
%J Comptes Rendus. Mathématique
%D 2008
%P 957-962
%V 346
%N 17-18
%I Elsevier
%R 10.1016/j.crma.2008.07.026
%G en
%F CRMATH_2008__346_17-18_957_0

Hoai-Minh Nguyen. Inequalities related to liftings and applications. Comptes Rendus. Mathématique, Volume 346 (2008) no. 17-18, pp. 957-962. doi : 10.1016/j.crma.2008.07.026. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.07.026/

References
Cited by

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

[2] J. Bourgain; H. Brezis; P. Mironescu $H^{\frac{1}{2}}$ maps with values into the circle: minimal connections, lifting, and the Ginzburg–Landau equation, Publ. Math. Inst. Hautes Etudes Sci., Volume 99 (2004), pp. 1-115

[3] J. Bourgain; H. Brezis; P. Mironescu Lifting, degree, and distributional Jacobian revisited, Comm. Pure Appl. Math., Volume 58 (2005), pp. 529-551

[4] J. Bourgain; H. Brezis; H.-M. Nguyen A new estimate for the topological degree, C. R. Acad. Sci. Paris, Ser. I, Volume 340 (2005), pp. 787-791

[5] H. Brezis; L. Nirenberg Degree theory and BMO, Part I: Compact manifolds without boundaries, Selecta Math., Volume 1 (1995), pp. 197-263

[6] R.R. Coifman; Y. Meyer Une généralisation du théorème de Calderon sur l'intégrale de Cauchy, Fourier Analysis, Proc. Sem., El Escorial, vol. 1, Asoc. Mat. Espa nola, Madrid, 1980, pp. 87-116

[7] P. Mironescu Sobolev maps on manifolds: degree, approximation, lifting, Perspectives in Nonlinear Partial Differential Equations, Contemp. Math., vol. 446, Amer. Math. Soc., Providence, RI, 2007, pp. 413-436 (In honor of Ham Brezis)

[8] P. Mironescu Lifting default for $S^{1}$ -valued maps, C. R. Acad. Sci. Paris, Ser. I, Volume 346 (2008) | DOI

[9] H.-M. Nguyen Optimal constant in a new estimate for the degree, J. d'Analyse Math., Volume 101 (2007), pp. 367-395

Cited by Sources:

Comments - Policy