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

Rivière, Tristan

Ann. Global Anal. Geom. 18, No. 5, 517-528 (2000).

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

Keywords:

density problems; Ginzburg-Landau functional; minimal connections; Sobolev maps between manifolds; Sobolev spaces; topological singularities; trace spaces