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.


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


Zbl 0657.46027
Full Text: DOI


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