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Lifting in Sobolev spaces. (English) Zbl 0967.46026
Any function \(u: \mathbb{R}^n\supset \Omega\to S^1\), \(S^1= \{z\in\mathbb{C}:|z|= 1\}\), can be written pointwise in a form \[ u(x)= e^{i\varphi(x)},\quad \varphi: \Omega\to \mathbb{R}.\tag{1} \] The lifting task is to find \(\varphi\) “as regular as \(u\) permits”. In the paper the lifting problem is regarded for the Sobolev spaces \(W^s_p(\Omega)\), \(0< s< \infty\), \(1< p<\infty\), \(\Omega\) is a (smooth) bounded domain in \(\mathbb{R}^n q\). The motivation for the question comes from variational problems of the Ginzburg-Landau type. More precisely, we say that the answer to the lifting problem is positive for the space \(W^s_p(\Omega, S^1)\) if every \(u\in W^s_p(\Omega, S^1)\) may be written in the form (1) for some \(\varphi\in W^s_p(\Omega, \mathbb{R})\). Otherwise, we say that the answer is negative.
It is proved that if \(n=1\) then the answer to the lifting question is always positive. If \(n\geq 2\) then it may be positive or negative. The full description is given in the paper. The answer is negative if \(1\leq s\) and \(sp< 2\), or \(s< 1\) and \(1\leq sp< n\). Otherwise, it is positive.
For the spaces \(W^{1/2}_s(\Omega, S^1)\) the answer is negative. Fortunately, the following inequality \[ \|\varphi\|_{W^s_p}\leq C(1- 2s)^{-1/2}\|u\|_{W^s_2}, \] holds, with \(C\) independent of \(u\) and \(s<{1\over 2}\). This allows to derive bounds for the Ginzburg-Landau functional.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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