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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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