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Approximation of Sobolev mappings. (English) Zbl 0820.46028

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 \(W^{1,k}(M^ m, S^ k)\), \(k\geq 2\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
58C25 Differentiable maps on manifolds

Citations:

Zbl 0547.58020
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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
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