## 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

### Keywords:

Sobolev maps; manifolds; embedding; Sobolev space

Zbl 0657.46027
Full Text:

### References:

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