Weak density of smooth maps for the Dirichlet energy between manifolds
Let \(M\) and \(N\) be compact smooth manifolds and assume that \(M\) is simply connected. In the main result of the article, the authors show that smooth maps are dense in the Sobolev space \(W^{1,2}(M,N)\) for the sequential weak topology. Moreover, if \(\varphi\) denotes an element of \(C^\infty (\partial M,N)\) smoothly extendable to \(M\), then every \(u\in W^{1,2} (M,N)\) satisfying \(u=\varphi\) on \(\partial M\) can be weakly approximated by a sequence of smooth maps \(u_m\) such that \(u_m=\varphi\) on \(\partial M\). Besides a delicate study of chains associated with the singular sets of mappings, the proof involves recent results of \textit{F. Hang} and \textit{F. Lin} [Math. Res. Lett. 8, No. 3, 321--330 (2001; Zbl 1049.46018)]. In the same spirit, the paper also contains several statements on the density of smooth maps in \(W^{1,p}({\mathbf B}^n,N)\).
scientific article; zbMATH DE number 1971830