Topology of sobolev mappings, II

Hang, Fengbo; Lin, Fanghua

doi:10.1007/bf02392696

Cited by 85 publications

(174 citation statements)

References 20 publications

(2 reference statements)

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“…Later, mainly due to the work of Bethuel, [5], it turned out that the condition for the density of smooth mappings can be formulated in terms of algebraic topology describing the topological structure of the manifolds M and N . Finally a necessary and sufficient condition for the density of smooth mappings in the space of Sobolev mappings W 1,p (M, N ) was discovered by Hang and Lin [29], [30]. This result corrects an earlier statement of Bethuel [5] about a necessary and sufficient condition for the density.…”

Section: Introductionsupporting

confidence: 74%

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Sobolev Mappings: Lipschitz Density is not a Bi-Lipschitz Invariant of the Target

Hajłasz

2007

GAFA, Geom. funct. anal.

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Abstract. We study a question of density of Lipschitz mappings in the Sobolev class of mappings from a closed manifold into a singular space. The main result of the paper, Theorem 1.7, shows that if we change the metric in the target space to a bi-Lipschitz equivalent one, than the property of the density of Lipschitz mappings may be lost. Other main results in the paper are Theorems 1.2, 1.3, 1.6, 1.8.

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Section: Introductionsupporting

confidence: 74%

“…This result corrects an earlier statement of Bethuel [5] about a necessary and sufficient condition for the density. Other papers on this topic include [4], [6], [7], [9], [10], [11], [12] [15], [16], [19], [20], [23], [24], [28], [29], [30], [34], [35], [50], [58], [59].…”

Section: Introductionmentioning

confidence: 99%

Sobolev Mappings: Lipschitz Density is not a Bi-Lipschitz Invariant of the Target

Hajłasz

2007

GAFA, Geom. funct. anal.

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“…In the general case of W 1,p (M, N ), 1 ≤ p < dim M, the necessary condition π [p] (N ) = 0 is not always sufficient for the density, see [18] for an example. The necessary and sufficient condition has been discovered by Hang and Lin in [19]. While we will not state this condition here, we will state a sufficient condition that was obtained earlier in [14], because this condition will play a role in what is to follow.…”

Section: Introductionmentioning

confidence: 91%

“…Bethuel [2] proved that in the local case (mappings from a ball) this condition is also sufficient. The proof of sufficiency is, however, very difficult (see [19] for corrections to Bethuel's paper). In the general case of W 1,p (M, N ), 1 ≤ p < dim M, the necessary condition π [p] (N ) = 0 is not always sufficient for the density, see [18] for an example.…”

Section: Introductionmentioning

confidence: 99%

Lipschitz homotopy and density of Lipschitz mappings in Sobolev spaces

Hajłasz¹,

Schikorra²

2014

Ann. Acad. Sci. Fenn. Math.

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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 (S n , Y ) are not dense in the Sobolev space W 1,n (S n , 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.

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“…The reader is warmly invited to consult the original paper [32] for the detailed discussions. Other types of results concerning S N -valued maps can be found in, e.g., [2,6,7,8,16,22,27,29,35,41,43,45,47,48,50,57,59,77,82] and the references therein.…”

Section: The Jacobian Distributional Of Maps From a Sphere Into Itselfmentioning

confidence: 99%

Estimates for the topological degree and related topics

Nguyên

2014

J. Fixed Point Theory Appl.

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Abstract. This is a survey paper on estimates for the topological degree and related topics which range from the characterizations of Sobolev spaces and BV functions to the Jacobian determinant and nonlocal filter problems in Image Processing. These results are obtained jointly with Bourgain and Brezis. Several open questions are mentioned.Mathematics Subject Classification. 55M25, 49J99, 46E35, 46B50.

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Topology of sobolev mappings, II

Cited by 85 publications

References 20 publications

Sobolev Mappings: Lipschitz Density is not a Bi-Lipschitz Invariant of the Target

Sobolev Mappings: Lipschitz Density is not a Bi-Lipschitz Invariant of the Target

Lipschitz homotopy and density of Lipschitz mappings in Sobolev spaces

Estimates for the topological degree and related topics

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