Lifting of BV functions with values in $$S^1$$.(English. Abridged French version)Zbl 1046.46026

Summary: We show that for every $$u\in BV(\Omega;S^1)$$, there exists a bounded variation functions $$\varphi\in BV(\Omega;\mathbb{R})$$ such that $$u=e^{i\varphi}$$ a.e. on $$\Omega$$ and $$|\varphi|_{BV}\leq 2| u|_{BV}$$. The constant 2 is optimal in dimension $$n>1$$.

MSC:

 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 26B30 Absolutely continuous real functions of several variables, functions of bounded variation
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References:

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