Density of smooth maps for fractional Sobolev spaces <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>W</mml:mi> <mml:mrow><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi></mml:mrow> </mml:msup></mml:math> into <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ℓ</mml:mi></mml:math> simply connected manifolds when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>s</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>

Bousquet, Pierre; Ponce, Augusto C.; Schaftingen, Jean Van

doi:10.5802/cml.5

Cited by 17 publications

(17 citation statements)

References 29 publications

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“…This approach has been proposed by Pakzad and Rivière [52], in the context of manifold-valued maps, in order to characterise strong limits of smooth N -valued maps in W 1,p (B d , N ). Because we are interested in vector-valued maps, our construction is different from theirs, and relies on the "projection trick" devised by Hardt, Kinderlehrer and Lin [38] (see also [36,19]). Eventually, we generalise Pakzad and Rivière's main result to a broader range of values for the exponent p, see Theorem 1 in Section 1.3.…”

Section: Introductionmentioning

confidence: 99%

Topological singular set of vector-valued maps, I: applications to manifold-constrained Sobolev and BV spaces

Canevari

Orlandi

2019

Calc. Var.

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We introduce an operator S on vector-valued maps u which has the ability to capture the relevant topological information carried by u. In particular, this operator is defined on maps that take values in a closed submanifold N of the Euclidean space R m , and coincides with the distributional Jacobian in case N is a sphere. More precisely, the range of S is a set of maps whose values are flat chains with coefficients in a suitable normed abelian group. In this paper, we use S to characterise strong limits of smooth, N -valued maps with respect to Sobolev norms, extending a result by Pakzad and Rivière. We also discuss applications to the study of manifold-valued maps of bounded variation. In a companion paper, we will consider applications to the asymptotic behaviour of minimisers of Ginzburg-Landau type functionals, with N -well potentials.

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Section: Introductionmentioning

confidence: 99%

Topological singular set of vector-valued maps, I: applications to manifold-constrained Sobolev and BV spaces

Canevari

Orlandi

2019

Calc. Var.

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“…Recall the criterion of [8] that, for k = 1, the vanishing of this "u topological singularity" chain (that is, the vanishing of such homotopy classes on a.e. such restriction about every a ∈ B m ) is equivalent the W k, p strong approximability of u by smooth maps (See also [10,11]). …”

Section: Topological Singularity and Bubblingmentioning

confidence: 96%

“…j (N ) = 0 for 0 ≤ j ≤ p − 1). (3) [11]: W k, p (B m , N ) with N being simply kp − 1 connected. (4) [26]: W 1,2 (B m , N ).…”

Section: Sequential Weak Approximationmentioning

confidence: 99%

Sequential weak approximation for maps of finite Hessian energy

Hardt

Rivière

2015

Calc. Var.

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Consider the space W 2,2 ( ; N ) of second order Sobolev mappings v from a smooth domain ⊂ R m to a compact Riemannian manifold N whose Hessian energy |∇ 2 v| 2 dx is finite. Here we are interested in relations between the topology of N and the W 2,2 strong or weak approximability of a W 2,2 map by a sequence of smooth maps from to N . We treat in detail W 2,2 (B 5 , S 3 ) where we establish the sequential weak W 2,2 density of W 2,2 (B 5 , S 3 ) ∩ C ∞ . The strong W 2,2 approximability of higher order Sobolev maps has been studied in the recent paper of Bousquet et al. (J. Eur. Math. Soc. (JEMS) 17 (4), 2015). For an individual map v ∈ W 2,2 (B 5 , S 3 ), we define a number L(v) which is approximately the total length required to connect the isolated singularities of a strong approximation u of v either to each other or to ∂ Bx 5 . Then L(v) = 0 if and only if v admits W 2,2 strongly approximable by smooth maps. Our critical result, obtained by constructing specific curves connecting the singularities of u, is the bound L(u) ≤ c + c B 5 |∇ 2 u| 2 dx . This allows us to construct, for the given Sobolev map v ∈ W 2,2 (B 5 , S 3 ), the desired W 2,2 weakly approximating sequence of smooth maps. To find suitable connecting curves for u, one uses the twisting of a u pull-back normal framing on a suitable level surface of u.Mathematics Subject Classification 56D15 · 46E35 · 46T20

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“…Many extensions, e.g. to more general Sobolev spaces, exists; see [17,30,28,6,8,7] and references within.…”

Section: H = W -Problem For Maps Into Manifolds and The Role Of Topologymentioning

confidence: 99%

Hölder-topology of the Heisenberg group

Schikorra

2020

Aequat. Math.

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The Heisenberg groups are examples of sub-Riemannian manifolds homeomorphic, but not diffeomorphic to the Euclidean space. Their metric is derived from curves which are only allowed to move in so-called horizontal directions.We report on some recent progress in the Analysis of the Hölder topology of the Heisenberg group, some related and some unrelated to density questions for Sobolev maps into the Heisenberg group.In particular we describe the main ideas behind a result by Haj lasz, Mirra, and the author regarding Gromov's conjecture, which is based on the linking number. We do not prove or disprove the Gromov Conjecture.

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Density of smooth maps for fractional Sobolev spaces W s,p into ℓ simply connected manifolds when s≥1

Cited by 17 publications

References 29 publications

Topological singular set of vector-valued maps, I: applications to manifold-constrained Sobolev and BV spaces

Topological singular set of vector-valued maps, I: applications to manifold-constrained Sobolev and BV spaces

Sequential weak approximation for maps of finite Hessian energy

Hölder-topology of the Heisenberg group

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