Low regularity function spaces of $N$-valued maps are contractible

Authors

  • Petru Mironescu

DOI:

https://doi.org/10.7146/math.scand.a-26360

Abstract

Let $M$ be a compact Lipschitz submanifold, possibly with boundary, of $\mathbb{R} ^n$. Let $N\subset \mathbb{R} ^k$ be an arbitrary set. Let $s\ge 0$ and $1\le p<\infty $ be such that $sp<1$. Then $W^{s, p}(M ; N)$ is contractible.

References

Adams, R. A., Sobolev spaces, Pure and Applied Mathematics, vol. 65, Academic Press, New York-London, 1975.

Bahri, A., Personal communication.

Brezis, H. and Mironescu, P., On some questions of topology for $S^1$-valued fractional Sobolev spaces, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 95 (2001), no. 1, 121–143.

Brezis, H. and Mironescu, P., Density in $W^{s,p}(Omega ;N)$, J. Funct. Anal. 269 (2015), no. 7, 2045–2109. https://doi.org/10.1016/j.jfa.2015.04.005

Brezis, H. and Nirenberg, L., Degree theory and BMO. I. Compact manifolds without boundaries, Selecta Math. (N.S.) 1 (1995), no. 2, 197–263. https://doi.org/10.1007/BF01671566

Magnot, J.-P., Remarks on the geometry and the topology of the loop spaces $H^s(S^1,N)$, for $sle 1/2$, preprint arXiv:1507.05772, 2015.

Triebel, H., Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Co., Amsterdam-New York, 1978.

Downloads

Published

2017-09-22

How to Cite

Mironescu, P. (2017). Low regularity function spaces of $N$-valued maps are contractible. MATHEMATICA SCANDINAVICA, 121(1), 144–150. https://doi.org/10.7146/math.scand.a-26360

Issue

Section

Articles