Let , , be such that . We prove that for each map one can find and such that . This yields a decomposition of u into a part that has a lifting in , , 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 , ). As a consequence, we find an intuitive proof for the existence of the distributional Jacobian Ju of maps (originally due to Bourgain, Brezis and the author). By completing a result of Bousquet, we characterize the distributions of the form Ju.
Soient , , tels que . Nous montrons que, pour chaque , il existe et tels que . Ceci donne une décomposition de u comme produit d'un facteur qui se relève dans , , 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 , ). Une conséquence de notre résultat est une preuve intuitive de l'existence du jacobien au sens des distributions Ju pour les applications (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.
Accepted:
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Petru Mironescu 1
@article{CRMATH_2010__348_13-14_743_0, author = {Petru Mironescu}, title = {Decomposition of $ {\mathbb{S}}^{1}$-valued maps in {Sobolev} spaces}, journal = {Comptes Rendus. Math\'ematique}, pages = {743--746}, publisher = {Elsevier}, volume = {348}, number = {13-14}, year = {2010}, doi = {10.1016/j.crma.2010.06.020}, language = {en}, }
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/
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