Characterization of function spaces via low regularity mollifiers

Smoothness of a function $f:{\mathbb R}^n\to {\mathbb R}$ can be measured in terms of the rate of convergence of $f\ast\rho_\varepsilon$ to $f$, where $\rho$ is an appropriate mollifier. In the framework of fractional Sobolev spaces, we characterize the "appropriate" mollifiers. We also obtain sufficient conditions, close to being necessary, which ensure that $\rho$ is adapted to a given scale of spaces. Finally, we examine in detail the case where $\rho$ is a characteristic function

Asymptotic behavior of critical points of an energy involving a loop-well potential

We describe the asymptotic behavior of critical points of $\int_{\Omega} [(1/2)|\nabla u|^2+W(u)/\varepsilon^2]$ when $\varepsilon\to 0$. Here, $W$ is a Ginzburg-Landau type potential, vanishing on a simple closed curve $\Gamma$. Unlike the case of the standard Ginzburg-Landau potential $W(u)=(1-|u|^2)^2/4$, studied by Bethuel, Brezis and H\'elein, we do not assume any symmetry on $W$ or $\Gamma$. In order to overcome the difficulties due to the lack of symmetry, we develop new tools which might be of independent interest

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

Existence of critical points with semi-stiff boundary conditions for singular perturbation problems in simply connected planar domains

Let $\Omega$ be a smooth bounded simply connected domain in $\mathbb{R}^2$. We investigate the existence of critical points of the energy $E_\varepsilon (u)=1/2\int_\Omega |\nabla u|^2+1/(4\varepsilon^2)\int_\Omega (1-|u|^2)^2$, where the complex map $u$ has modulus one and prescribed degree $d$ on the boundary. Under suitable nondegeneracy assumptions on $\Omega$, we prove existence of critical points for small $\varepsilon$. More can be said when the prescribed degree equals one. First, we obtain existence of critical points in domains close to a disc. Next, we prove that critical points exist in "most" of the domains

Superposition with subunitary powers in Sobolev spaces

International audienceLet $0$<$a$<$1$ and set $\Phi (t)=|t|^a$, $t\in {\mathbb R}$. We prove that the superposition operator $u\mapsto \Phi (u)$ maps the Sobolev space $W^{1,p}({\mathbb R}^n)$ into the fractional Sobolev space $W^{a,p/a}({\mathbb R}^n)$. We also investigate the case of more general nonlinearities

S1{\mathbb S}^1-valued Sobolev maps

International audienceWe describe the structure of the space $W^{s,p}({\mathbb S}^n ; {\mathbb S}^1)$, where $0$<$s$<$\infty$ and $1\le p$<$\infty$. According to the values of $s$, $p$ and $n$,maps in $W^{s,p}({\mathbb S}^n ; {\mathbb S}^1)$ can either be characterised by their phases, or by a couple (singular set, phase). Here are two examples:a) $W^{1/2,6}({\mathbb S}^3 ; {\mathbb S}^1) = \{e^{\imath\varphi};\, \varphi\in W^{1/2,6} + W^{1,3}\}$;b) $W^{1/2,3}( {\mathbb S}^2 ; {\mathbb S}^1) \approx D \times \{e^{\imath \varphi} ;\, \varphi\in W^{1/2,3} + W^{1,3/2}\}$. In the second example, $D$ is an appropriate set of infinite sums of Dirac masses. The sense of "$\approx$" will be explained in the paper. The presentation is based on a paper of H.-M. Nguyen (C. R. Acad. Sci. Paris 2008), and on a forthcoming paper of the author

Low regularity function spaces of N-valued maps are contractible

International audienceLet $M$ be a compact Lipschitz submanifold, possibly with boundary, of ${\mathbb R}^n$. Let $N\subset {\mathbb R}^k$ be an arbitrary set. Let $s\ge 0$ and $1\le p<\infty$ be such that $sp<1$. Then $W^{s, p}(M ; N)$ is contractible

On some inequalities of Bourgain, Brezis, Maz'ya, and Shaposhnikova related to L1L^1 vector fields

International audienceBourgain and Brezis (J. Amer. Math. Soc. 2003) established, for maps $f\in L^n({\mathbb T}^n)$ with zero average, the existence of a solution $\vec{Y}\in W^{1,n}\cap L^\infty$ of (1) div $\vec{Y}=f$. Maz'ya (Contemp. Math. vol. 445) proved that if, in addition $f\in H^{n/2-1}({\mathbb T}^n)$, then (1) can be solved in $H^{n/2}\cap L^\infty$. Their arguments are quite different. We present an elementary property of the biharmonic operator in two dimensions. This property unifies, in two dimensions, the two approaches, and implies another (apparently unrelated) estimate of Maz'ya and Shaposhnikova (Sobolev spaces in mathematics I, 2009). We discuss higher dimensional analogs of the above results

Lifting default for S1{\mathbb S}^1-valued maps

International audienceLet $\varphi\in C^\infty ([0,1]^N ; {\mathbb R})$ and set $g=e^{\imath\varphi}$. When $0$$1$ and $sp\ge 1$, or when $N\ge 2$, $p$>$1$ and $sp$>$1$. In this work, we establish the existence of a control in the full range $0$<$s$<$1$, $p\ge 1$ and $N\ge 1$. In particular, we do not require that $sp\ge 1$, as in the previous results

Decomposition of S1{\mathbb S}^1-valued maps in Sobolev spaces

International audienceLet $n\ge 2$, $s$>$0$ and $p\ge 1$ be such that $1\le sp$<$2$. We prove that for each map $u\in W^{s,p}({\mathbb S}^n ; {\mathbb S}^1)$ one can find some $\varphi\in W^{s,p}({\mathbb S}^n ; {\mathbb R})$ and some $v\in W^{sp, 1}({\mathbb S}^n ; {\mathbb S}^1)$ such that $u=e^{\imath\varphi}\, v$. This yields a decomposition of $u$ into a part, $e^{\imath\varphi}$, that has a lifting in $W^{s,p}$, and a map, $v$, "smoother" than $u$ but which need not have a lifting within $W^{s,p}$. Our result generalizes a previous one of Bourgain and Brezis (J. Amer. Math. Soc. 2003), which corresponds to $s=1/2$ and $p=2$. As a consequence of the above factorization $u=e^{\imath\varphi}\, v$, we find an intuitive proof of the existence of the Jacobian $J u$ of maps $u\in W^{s, p}({\mathbb S}^n ; {\mathbb S}^1)$, result originally due to Bourgain, Brezis and the author (Comm. Pure Appl. Math. 2005). By completing a result of Bousquet (J. Anal. Math. 2007), we characterize the distributions of the form $J u$