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

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• 作者：Petru Mironescu ^{mironescu@math.univ-lyon1.fr" class="auth_mail" title="E-mail the corresponding author}
• 刊名：Comptes Rendus Mathematique
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：353
• 期：12
• 页码：1087-1092
• 全文大小：286 K

文摘

When 1<p<∞$1<p<\infty$, maps f   in W^1/p,p((0,1);S¹)$W^{1/p,p}((0,1);{\mathbb{S}}^1)$ have 17a7ecd7b311" title="Click to view the MathML source">W^1/p,p$W^{1/p,p}$ phases φ  , but the 17a7ecd7b311" title="Click to view the MathML source">W^1/p,p$W^{1/p,p}$-seminorm of φ is not controlled by the one of f  . Lack of control is illustrated by “the kink”: f=e^ıφ$f={\mathrm{e}}^{ı\phi }$, where the phase φ moves quickly from 0 to 2π  . A similar situation occurs for maps f:S¹→S¹$f:{\mathbb{S}}^1\to {\mathbb{S}}^1$, with Moebius maps playing the role of kinks. We prove that this is the only loss of control mechanism: each map f:S¹→S¹$f:{\mathbb{S}}^1\to {\mathbb{S}}^1$ satisfying ${|f|}_{W^{1/p,p}}^p\le M$ can be written as $f={\mathrm{e}}^{ı\psi }\prod _{j=1}^K{(M_{a_j})}^{\pm 1}$, where M_aj$M_{a_j}$ is a Moebius map vanishing at a_j∈D$a_j\in \mathbb{D}$, while the integer K=K(f)$K=K(f)$ and the phase ψ are controlled by M  . In particular, we have $K\le c_{p}M$ for some c_p$c_p$. When p=2$p=2$, we obtain the sharp value of c₂$c_2$, which is c₂=1/(4π²)$c_2=1/(4{\pi }^2)$. As an application, we obtain the existence of minimal maps of degree one in a708d7aa5854" title="Click to view the MathML source">W^1/p,p(S¹;S¹)$W^{1/p,p}({\mathbb{S}}^1;{\mathbb{S}}^1)$ with p∈(2−ε,2)$p\in (2-\epsilon ,2)$.