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\(W^{1,1}\)-maps with values into \(S^1\). (English) Zbl 1078.46020

Chanillo, Sagun (ed.) et al., Geometric analysis of PDE and several complex variables. Dedicated to François Trèves. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3386-3/pbk). Contemporary Mathematics 368, 69-100 (2005).
Let \(G\subset\mathbb{R}^3\) be a smooth bounded domain with \(\Omega= \partial G\) simply connected. Set \[ W^{1,1}(\Omega, S^1):= \{g\in W^{1,1}(\Omega, \mathbb{R}^2)\mid|g|= 1\text{ a.e. on }\Omega\}. \] In this paper, the authors deal with the following questions:
1. Properties of \(W^{1,1}(S^1, S^1)\).
2. Properties of \(W^{1,1}(\Omega, S^1)\).
3. \(W^{1,1}(\Omega, S^1)\) and relaxed Jacobians.
4. Further directions and open problems.
For the entire collection see [Zbl 1058.35003].

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
58D15 Manifolds of mappings
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