Density in Ws,p(Ω;N)W^{s,p}(\Omega ; N)

Let $\Omega$ be a smooth bounded domain in ${\mathbb R}^n$, 0\textless{}s\textless{}\infty and 1\le p\textless{}\infty. We prove that $C^\infty(\overline\Omega\, ; {\mathbb S}^1)$ is dense in $W^{s,p}(\Omega ; {\mathbb S}^1)$ except when 1\le sp\textless{}2 and $n\ge 2$. The main ingredient is a new approximation method for $W^{s,p}$-maps when s\textless{}1. With 0\textless{}s\textless{}1, 1\le p\textless{}\infty and sp\textless{}n, $\Omega$ a ball, and $N$ a general compact connected manifold, we prove that $C^\infty(\overline\Omega \, ; N)$ is dense in $W^{s,p}(\Omega \, ; N)$ if and only if $\pi\_{[sp]}(N)=0$. This supplements analogous results obtained by Bethuel when $s=1$, and by Bousquet, Ponce and Van Schaftingen when $s=2,3,\ldots$ [General domains $\Omega$ have been treated by Hang and Lin when $s=1$; our approach allows to extend their result to s\textless{}1.] The case where s\textgreater{}1, $s\not\in{\mathbb N}$, is still open.Comment: To appear in J. Funct. Anal. 49

Regularity of minimizers of relaxed problems for harmonic maps

AbstractWe prove that every minimizer on H1(Ω; S2) of the relaxed energy ∝¦▽u¦2 + 8πλL(u), where 0 ⩽ λ < 1 and L(u) is the length of a minimal connection connecting the singularities of u, is smooth except at a finite number of points

On some questions of topology for S1{\mathbb S}^1-valued fractional Sobolev spaces

International audienceWe describe the homotopy classes of the metric space $W^{s,p}(U ; {\mathbb S}^1)$, where $U$ is an open set in ${\mathbb R}^n$

Lifting in Sobolev spaces

International audienceWe characterize the couples $(s,p)$ with the following property: if $u$ is a complex-valued unimodular map in $W^{s,p}$, then $u$ has (locally) a phase in $W^{s,p}$

On a semilinear elliptic equation with inverse square potential

On a semilinear elliptic equation with inverse square potentia