Lifting of ${\mathbb S}^1$-valued maps in sums of Sobolev spaces - Archive ouverte HAL Access content directly
Preprints, Working Papers, ... Year : 2008

Lifting of ${\mathbb S}^1$-valued maps in sums of Sobolev spaces

Abstract

We describe, in terms of lifting, the closure of smooth ${\mathbb S}^1$-valued maps in the space $W^{s,p} ((−1, 1)^N ; {\mathbb S}^1)$. Here, $0$<$s$<$\infty$ and $1\le p$<$\infty$. This description follows from an estimate for the phase of smooth maps: let $0$<$s$<$1$, $\varphi\in C^\infty ([-1,1]^N ; {\mathbb R})$ and set $u=e^{\imath\varphi}$. Then we may split $\varphi=\varphi_1+\varphi_2$, where the smooth maps $\varphi_1$ and $\varphi_2$ satisfy (*) $|\varphi_1|_{W^{s,p}}\le C\, |u|_{W^{s,p}}$ and $\|\nabla\varphi_2\|_{L^{sp}}\le C\, |u|_{W^{s,p}}^{1/s}$. Estimate (*) was obtained by Bourgain and Brezis (J. Amer. Math. Soc. 2003) when $s=1/2$ and $p=2$, and by Nguyen (C. R. Acad. Sci. 2008) when $N=1$, $p$>$1$ and $s=1/p$. Our proof is a sort of continuous version of the Bourgain-Brezis approach (based on paraproducts). Estimate (∗) answers (and generalizes) a question of Bourgain, Brezis and the author (Comm. Pure Appl. Math. 2005).
Fichier principal
Vignette du fichier
lifting_sobolev_sums_20080623.pdf (384.72 Ko) Télécharger le fichier
Origin : Files produced by the author(s)
Loading...

Dates and versions

hal-00747663 , version 1 (31-10-2012)

Identifiers

  • HAL Id : hal-00747663 , version 1

Cite

Petru Mironescu. Lifting of ${\mathbb S}^1$-valued maps in sums of Sobolev spaces. 2008. ⟨hal-00747663⟩
192 View
86 Download

Share

Gmail Facebook X LinkedIn More