## 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:

 [1] [Be1]Bethuel, F., A characterization of maps inH 1(B 3,S 2) which can be approximated by smooth maps. To appear. [2] [BBC]Bethuel, F., Brezis, H. & Coron, J. M., Relaxed energies for harmonic maps. To appear. [3] [BZ]Bethuel, F. &Zheng, X., Density of smooth functions between two manifolds in Sobolev spaces.J. Funct. Anal. 80 (1988), 60–75. · Zbl 0657.46027 [4] [BCL]Brezis, H., Coron, J. M. &Lieb, E., Harmonic maps with defects.Comm. Math. Phys., 107 (1986), 649–705. · Zbl 0608.58016 [5] [CG]Coron, J. M. & Gulliver, R., Minimizingp-harmonic maps into spheres. To appear. [6] [EL]Eells, J. &Lemaire, L., A report on harmonic maps.Bull. London Math. Soc., 10 (1978), 1–68. · Zbl 0401.58003 [7] [E]Escobedo, M. To appear. [8] [F]Fuchs, M.,p-harmonic Hindernisprobleme. Habilitationsschrift, Düsseldorf, 1987. [9] [HL]Hardt, R. &Lin, F. H., Mappings minimizing theL p norm of the gradient.Comm. Pure Appl. Math., 40 (1987), 556–588. · Zbl 0646.49007 [10] [H]Helein, F., Approximations of Sobolev maps between an open set and an Euclidean sphere, boundary data, and singularities. To appear. [11] [KW]Karcher, M. &Wood, J. C., Non existence results and growth properties for harmonic maps and forms.J. Reine Angew. Math., 353 (1984), 165–180. · Zbl 0544.58008 [12] [L]Luckhaus, S., Partial Hölder continuity of minima of certain energies among maps into a Riemannian manifold. To appear inIndiana Univ. Math. J. · Zbl 0641.58012 [13] [SU1]Schoen, R. &Uhlenbeck, K., A regularity theory for harmonic maps.J. Differential Geom., 17 (1982), 307–335. · Zbl 0521.58021 [14] [SU2]–, Boundary regularity and the Dirichlet problem for harmonic maps.J. Differential Geom., 18 (1983), 253–268. · Zbl 0547.58020 [15] [SU3]Schoen, R. & Uhlenbeck, K., Approximation theorems for Sobolev mappings. To appear. [16] [W1]White, B., Infima of energy functionals in homotopy classes.J. Differential Geom., 23 (1986), 127–142. · Zbl 0588.58017 [17] [W2]–, Homotopy classes in Sobolev spaces and the existence of energy minimizing maps.Acta Math., 160 (1988), 1–17. · Zbl 0647.58016 [18] [Wo]Wood, J. C., Non existence of solution to certain Dirichlet problems for harmonic maps. Preprint Leeds University (1981).
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.