Hajłasz, Piotr Approximation of Sobolev mappings. (English) Zbl 0820.46028 Nonlinear Anal., Theory Methods Appl. 22, No. 12, 1579-1591 (1994). Let \(M^ n\) and \(N^ n\) be two compact Riemannian manifolds, \(\partial N^ n= \varnothing\) and assume that \(N^ n\) is embedded in \(\mathbb{R}^ \nu\). The space \(W^{1,p}(M^ m, N^ n)\) is defined by \[ W^{1,p}(M^ m, N^ n)= \bigl\{ f\in W^{1,p} (M^ m, \mathbb{R}^ \nu);\;f(x)\in N^ n\text{ a.e. } x\in M^ m\},\quad 1\leq p< \infty. \] The following theorem is proved: If \(\pi_ 1(N^ n)=\cdots= \pi_ k(N^ n)= 0\) \((k\in \mathbb{N}^*)\) and:a) \(1\leq p< k+1\), then \(C^ \infty(M^ m, N^ n)\) is dense in \(W^{1,p}(M^ m, N^ n)\) in the norm topology;b) \(p= k+1\), then \(C^ \infty(M^ m, N^ n)\) is sequentially dense for the weak topology in \(W^{1,p}(M^ m, N^ n)\).It was known that if \(p\geq m\), then \(C^ \infty(M^ m, N^ n)\) is dense in \(W^{1,p}(M^ m, N^ n)\) [see R. Schoen and K. Uhlenbeck, J. Differ. Geom. 18, 253-268 (1983; Zbl 0547.58020)].The techniques used by the author in the proof of the above theorem are triangulations, skeletons, retractions and Lipschitz mappings.As a corollary, it is shown that \(C^ \infty(M^ m, S^ k)\) is sequentially dense for the weak topology in \(W^{1,k}(M^ m, S^ k)\), \(k\geq 2\). Reviewer: I.Mihai (Bucureşti) Cited in 2 ReviewsCited in 21 Documents MSC: 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 58C25 Differentiable maps on manifolds Keywords:Sobolev mappings; Riemannian manifolds; triangulations; skeletons; retractions; Lipschitz mappings Citations:Zbl 0547.58020 PDFBibTeX XMLCite \textit{P. Hajłasz}, Nonlinear Anal., Theory Methods Appl. 22, No. 12, 1579--1591 (1994; Zbl 0820.46028) Full Text: DOI References: [2] Schoen, R.; Uhlenbeck, K., The Dirichlet problem for harmonic maps, J. diff. Geom., 18, 153-268 (1983) · Zbl 0547.58020 [4] Bethuel, F., The approximation problem for Sobolev maps between two manifolds, Acta math., 167, 153-206 (1991) · Zbl 0756.46017 [5] Bojarski, B., Geometric properties of Sobolev mappings, Pitman Research Notes in Mathematics, Vol. 211, 225-241 (1989) · Zbl 0689.46014 [6] Borsuk, K., Quelques relations entre la situation des ensambles et la retraction dans les espaces euclidiens, Fund. Math., 29, 191-205 (1937) · JFM 63.1161.01 [7] Bethuel, F.; Zheng, X., Density of smooth functions between two manifolds in Sobolev spaces, J. funct. Analysis, 80, 60-75 (1988) · Zbl 0657.46027 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.