Lipschitz homotopy and density of Lipschitz mappings in Sobolev spaces

2013/06/27 by Piotr Hajlasz, Armin Schikorra, Hajlasz, Piotr +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Topology (math.GT) #Primary 46E35 #Secondary 55Q70 #math.FA #math.GT #msc:46E35 #msc:55Q70

paper · pdf · doi:10.48550/arxiv.1306.6502

arxiv created 2013/06/27 · arxiv updated 2013/06/28

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-Poincare inequality.

Related

• On the lack of density of Lipschitz mappings in Sobolev spaces with Heisenberg target
• Examples and applications of the density of strongly norm attaining Lipschitz maps
• Sobolev extensions of Lipschitz mappings into metric spaces
• Yet another proof of the density in energy of Lipschitz functions
• Biduality and density in Lipschitz function spaces