Profile decomposition and phase control for circle-valued maps in one dimension
Résumé
When $p\in (1,\infty)$, maps $f$ in $W^{1/p,p}((0,1) ; {\mathbb S}^1)$ have $W^{1/p,p}$ phases $\varphi$, but the $W^{1/p,p}$-seminorm of $\varphi$ is not controlled by the one of $f$. Lack of control is illustrated by ‘’the kink’’: $f = e^{i\varphi}$, where the phase $\varphi$ moves quickly from $0$ to $2\pi$. A similar situation occurs for maps $f:{\mathbb S}^1\to{\mathbb S}^1$, with Moebius maps playing the role of kinks. We prove that this is the only loss of control mechanism. As an application, we obtain the existence of minimal maps of degree one in $W^{1/p,p}({\mathbb S}^1 ; {\mathbb S}^1)$ with $p\in (2-\varepsilon,2)$.
Domaines
Analyse classique [math.CA]
Origine : Fichiers produits par l'(les) auteur(s)
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