## $$W^{1,1}$$-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

### Keywords:

optimal lifting; topological obstacle; geodesics