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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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