Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 39, 2014, 593-604

LIPSCHITZ HOMOTOPY AND DENSITY OF LIPSCHITZ MAPPINGS IN SOBOLEV SPACES

Piotr Hajlasz and Armin Schikorra

University of Pittsburgh, Department of Mathematics
301 Thackeray Hall, Pittsburgh, PA 15260, U.S.A.; hajlasz 'at' pitt.edu

Max-Planck Institut MiS Leipzig
Inselstr. 22, 04103 Leipzig, Germany; armin.schikorra 'at' mis.mpg.de

Abstract. We construct a smooth compact n-dimensional manifold Y with one point singularity such that all its Lipschitz homotopy groups are trivial, but Lipschitz mappings Lip(Sⁿ,Y) are not dense in the Sobolev space W^1,n(Sⁿ,Y). On the other hand we show that if a metric space Y is Lipschitz (n - 1)-connected, then Lipschitz mappings Lip(X,Y) are dense in N^1,p(X,Y) whenever the Nagata dimension of X is bounded by n and the space X supports the p-Poincaré inequality. Lipschitz (n - 1)-connectedness is a stronger condition than vanishing of the first n - 1 Lipschitz homotopy groups as it assumes quantitative estimates of Lipschitz constants.

2010 Mathematics Subject Classification: Primary 46E35; Secondary 55Q70.

Key words: Sobolev mappings, density, Lipschitz homotopy groups, metric spaces.

Reference to this article: P. Hajlasz and A. Schikorra: Lipschitz homotopy and density of Lipschitz mappings in Sobolev spaces. Ann. Acad. Sci. Fenn. Math. 39 (2014), 593-604.

Full document as PDF file

doi:10.5186/aasfm.2014.3932