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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:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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