Approximations of Sobolev maps between an open set and an Euclidean sphere, boundary data, and singularities (Q1112233)

We study density of smooth maps between an open subset \(\Omega\) of \({\mathbb{R}}^ N\) and an euclidean sphere \(S^{M-1}\) in \(W^{1,p}(\Omega,S^{M-1})\). We show that given a closed subset \({\mathcal S}\) of \({\mathbb{R}}^ n\) contained in \(\Omega\) with a Hausdorff measure of dimension \((N-p)\) zero, and given a map \(f\) in \(W^{1,p}(\Omega,S^{M-1})\) which is smooth on \(\Omega\setminus {\mathcal S}\), then the set of maps from \(\Omega\) to \(S^{M-1}\) which are smooth on \(\Omega\setminus {\mathcal S}\) and which agree with f outside a compact subset of \(\Omega\setminus {\mathcal S}\) is dense in \(G(f)=\{u\in W^{1,p}(\Omega,S^{M- 1})\), \(f|_{\partial \Omega}=u|_{\partial \Omega}\}\) provided that \(1\leq p<M-1.\) We apply this result to show that the map \(u_*(x)=x/| x|\) from the unit ball of \({\mathbb{R}}^ N\) to \(S^{N-1}\) minimizes ``strongly'' the quadratic energy functional when \(N\) is greater or equal to 9.