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The approximation problem for Sobolev maps between two manifolds. (English) Zbl 0756.46017

Let \(M^ n\) and \(N^ k\) be two compact Riemannian manifolds of dimensions \(n\) and \(k\) respectively. \(N^ k\) is isometrically embedded in \(R^ \ell\) (\(\ell\in N^*\)). \(M^ n\) may have a boundary but not \(N^ k\). For \(1\leq p\leq n\) the Sobolev space \(W^{1,p}(M^ n,N^ k)\) is defined by \[ W^{1,p}(M^ n,N^ k)=\{u\in W^{1,p}(M^ n,R^ \ell);\;u(x)\in N^ k\text{ a.e.}\}. \] With these preliminaries the main result of this work asserts:
Let \(1\leq p\leq n\). Smooth maps between \(M^ n\) and \(N^ k\) are dense in \(W^{1,p}(M^ n,N^ k)\) if and only if \(\pi_{[p]}(N^ k)=0\). (Here \([p]\) represents the largest integer less than or equal to \(p\) and \(\pi\) is the radial projection).
Necessity has already been proved by F. Bethuel and X. Zheng [J. Funct. Anal. 80, 60-75 (1988; Zbl 0657.46027)], much of the paper is devoted to prove the sufficiency.
When \(\pi_{[p]}(N^ k)\neq 0\), this theorem is of little use. Even in that case maps in \(W^{1,p}(M^ n,N^ k)\) may be approximated by maps which are regular except on a simple set of low dimension. A class \(R_ p^ 0\) (resp. \(R_ p^ \infty\)) of maps in \(W^{1,p}(M^ n,N^ k)\) is defined in the following way: \(u\in W^{1,p}(M^ n,N^ k)\) is in \(R_ p^ 0\) (resp. \(R_ p^ \infty\)) if and only if \(u\) is continuous (resp. smooth) except on a singular set \(\Sigma(u)\), where \(\Sigma(u)=\bigcup_{j=1}^ r \Sigma_ i\), \(r\in N^*\), where \(i=1,\dots,r\), \(\Sigma_ i\) is smooth; if \(p>n-1\), \(\Sigma_ i\) is a point. The author proves that
For every \(1\leq p\leq n\) \(R_ p^ 0\) (resp. \(R_ p^ \infty\)) is dense in \(W^{1,p}(M^ n,N^ k)\).
For the case \(\pi_{[p]}(N^ k)\neq 0\), the problem of density of smooth maps for the weak topology is als considered and a number of interesting theorems are proved.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
58D15 Manifolds of mappings
54C35 Function spaces in general topology

Citations:

Zbl 0657.46027
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References:

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