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Journal Articles Comptes Rendus. Mathématique Year : 2015

Profile decomposition and phase control for circle-valued maps in one dimension

Abstract

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 $p\in (2-\varepsilon,2)$.
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Dates and versions

hal-01202156 , version 1 (22-09-2015)
hal-01202156 , version 2 (03-01-2016)

Identifiers

  • HAL Id : hal-01202156 , version 2

Cite

Petru Mironescu. Profile decomposition and phase control for circle-valued maps in one dimension. Comptes Rendus. Mathématique, 2015, 353 (12), pp.1087-1092. ⟨hal-01202156v2⟩
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