## Dense subsets of $$H^{1/2}(S^2,S^1)$$.(English)Zbl 0960.35022

Summary: We prove that the maps from $$S^2$$ into $$S^1$$ having a finite number of isolated singularities of degree $$\pm 1$$ are dense for the strong topology in $$H^{1/2}(S^2, S^1)$$. We also prove that smooth maps are dense in $$H^{1/2}(S^2, S^1)$$ for the sequentially weak topology and that this is not the case in $$H^s(S^2, S^1)$$ for $$s> 1/2$$.

### MSC:

 35J20 Variational methods for second-order elliptic equations 35J67 Boundary values of solutions to elliptic equations and elliptic systems 35J60 Nonlinear elliptic equations 58-XX Global analysis, analysis on manifolds 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 46E40 Spaces of vector- and operator-valued functions
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