Density of Lipschitz mappings in the class of Sobolev mappings between metric spaces
type: Item
instance of: scholarly article
title: Density of Lipschitz mappings in the class of Sobolev mappings between metric spaces
zbMATH Open document ID: 1175.46022
author: Piotr Hajłasz
published in: Mathematische Annalen
publication date: 2009-03-26T00:00:00Z
review text: Let \((X,d,\mu)\) be a metric-measure space equipped with a finite doubling measure. The space \((X,d,\mu)\) is said to support the \(p\)-Poincaré inequality for some \(p\in[1,\infty)\) if \(\mu\) is a doubling measure and there exist constants \(C_p >0\) and \(\lambda\geq1\) such that for every ball \(B\subset X\), for every integrable function \(f\in L^1(\lambda B)\) and for every \(0\leq g\in L^p (\lambda B)\) being a \(p\)-weak upper gradient of \(f\) on \(\lambda B\), the inequality \[ \frac{1}{\mu(B)}\int_B |f-f_B|\,d\mu\leq C_p\,({\mathrm{\,diam\,}}B) \left(\frac{1} {\mu(\lambda B)}\int_{\lambda B} g^p \,d\mu\right)^{1/p} \] is satisfied, where \(f_B=\frac{1} {\mu(B)}\int_B f \,d\mu\). Also, for \(p\in[1,\infty)\), any metric-measure space \((X,d,\mu)\) and metric space \((Y,d_Y)\), the mapping \(F\in L^p (X,Y)\) is said to belong to the Newtonian-Sobolev class of mappings \(N^{1,p}(X,Y)\) if there is a Borel function \(0\leq g\in L^p (X)\) such that \[ d_Y (F(\gamma(a)),F(\gamma(b)))\leq\int_{\gamma}g \] for every rectifiable curve \(\gamma : [a,b]\rightarrow X\). In this paper, the author proves that Lipschitz mappings are dense in the Newton--Sobolev classes \(N^{1,p}(X,\,Y)\) of mappings from spaces \(X\) supporting \(p\)-Poincaré inequalities into a finite Lipschitz polyhedron \(Y\) if and only if \(Y\) is \([p]\)-connected, \(\pi_1 (Y)=\pi_2 (Y)=\cdots=\pi_{[p]}(Y)=0\), where \(p\in[1,\infty)\) and \([p]\) denotes the largest integer less than or equal to \(p\).
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 5533127
zbMATH Keywords: metric space
zbMATH Keywords: Lipschitz polyhedron
zbMATH Keywords: Lipschitz mapping
zbMATH Keywords: density
zbMATH Keywords: \(p\)-Poincaré inequality
zbMATH Keywords: Newtonian-Sobolev space
reviewed by: Dachun Yang
MaRDI profile type: Publication
full work available at URL: s00208-008-0291-7
OpenAlex ID: W2160831105
cites work: Q3211883
cites work: A characterization of maps in \(H^ 1(B^ 3,S^ 2)\) which can be approximated by smooth maps
cites work: The approximation problem for Sobolev maps between two manifolds
cites work: Density of smooth functions between two manifolds in Sobolev spaces
cites work: Q5302131
cites work: Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings
cites work: Topology and Sobolev spaces
cites work: Degree theory of BMO. I: Compact manifolds without boundaries
cites work: Degree theory and BMO. II: Compact manifolds with boundaries. (Appendix with Petru Mironescu)
cites work: Legendrian energy minimizers. I: Heisenberg group target
cites work: Differentiability of Lipschitz functions on metric measure spaces
cites work: Minimizing p-harmonic maps into spheres.
cites work: Q3468152
cites work: Harmonic maps between Riemannian polyhedra. With a preface by M. Gromov
cites work: A Report on Harmonic Maps
cites work: Q3834525
cites work: Definitions of Sobolev classes on metric spaces
cites work: Harmonic maps into singular spaces and \(p\)-adic superrigidity for lattices in groups of rank one
cites work: Equivalent statement of the Poincaré conjecture
cites work: Approximation of sobolev mappings
cites work: Sobolev spaces on an arbitrary metric space
cites work: Q4453425
cites work: Sobolev mappings: Lipschitz density is not a bi-Lipschitz invariant of the target
cites work: Weakly differentiable mappings between manifolds
cites work: Sobolev met Poincaré
cites work: Density problems for W1,1 (M, N)
cites work: On the weak limits of smooth maps for the Dirichlet energy between manifolds
cites work: Topology of Sobolev mappings
cites work: Topology of Sobolev mappings. II.
cites work: Topology of Sobolev mappings III
cites work: An 𝑛-dimensional space that admits a Poincaré inequality but has no manifold points
cites work: Quasiconformal maps in metric spaces with controlled geometry
cites work: Sobolev classes of Banach space-valued functions and quasiconformal mappings
cites work: Characterization of the strong closure of \(C^ \infty(B^ 4;S^ 2)\) in \(W^{1,p}(B^ 4;S^ 2)\) (\({16\over 5}\leq p < 4\))
cites work: Equilibrium maps between metric spaces
cites work: Generalized Dirichlet forms and harmonic maps
cites work: Q4232746
cites work: Q4944263
cites work: The Poincaré inequality is an open ended condition
cites work: Sobolev spaces and harmonic maps for metric space targets
cites work: Some function spaces on spaces of homogeneous type
cites work: Ahlfors \(Q\)-regular spaces with arbitrary \(Q>1\) admitting weak Poincaré inequality
cites work: Extending Lipschitz functions via random metric partitions
cites work: Weak density of smooth maps in \(W^{1, 1}(M,N)\) for non-Abelian \({\pi}_1(N)\)
cites work: Weak density of smooth maps for the Dirichlet energy between manifolds
cites work: Sobolev-type classes of functions with values in a metric space
cites work: Newtonian spaces: An extension of Sobolev spaces to metric measure spaces
cites work: Boundary regularity and the Dirichlet problem for harmonic maps
cites work: Boundary regularity of harmonic maps to nonpositively curved metric spaces
cites work: Local minimizers and quasiconvexity -- the impact of topology
cites work: Q4328597
cites work: Infima of energy functionals in homotopy classes of mappings
cites work: Homotopy classes in Sobolev spaces and the existence of energy minimizing maps
cites work: Weakly Differentiable Functions
DOI: 10.1007/S00208-008-0291-7
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