Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

Analytical obstructions to the weak approximation of Sobolev mappings into manifolds

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For every integer $p\ge 2$ there is a compact manifold target into which some Sobolev map cannot be weakly approximated by smooth maps.

desk verdict Strong paper proving analytical obstructions for every integer p≥2, but the higher-order sphere extension (Thm 1.4) is stated without proof and should be either supplied or downgraded. read the letter →

arxiv 2412.12889 v2 pith:JMBGZS33 submitted 2024-12-17 math.FA math.AP

Antoine Detaille , Jean Van Schaftingen This is my paper · ORCID
classification math.FAmath.AP MSC 58D1546E3546T1058C25
keywords weakapproximationofSobolevmappingsrelaxedenergyanalyticalobstructionBrouwerdegreeestimateWhiteheadproductHopfinvariantbubblingmapshigher-orderspaces
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes that weak approximation by smooth maps fails for every integer exponent $p\ge 2$. Specifically, for each such $p$ it constructs a compact manifold $N$ (retracting onto the $p$-skeleton of a $(p+1)$-torus, and for a parameter choice just a connected sum of two $(p+1)$-tori) such that whenever the domain dimension exceeds $p$, there is a finite-energy map $M\to N$ that is not the weak limit of any sequence of smooth maps. The obstruction is analytical, not topological: it is produced by a periodic singular retraction whose relaxed energy grows like $\ell^{p+1}\ln\ell$ on cubes of side $\ell$, while its ordinary Sobolev energy only grows like $\ell^{p+1}$. For $p=4n-1$ the same phenomenon occurs with the sphere $S^{2n}$ as the target, generalizing the previously known $p=3$ case. The same results are obtained for higher-order Sobolev spaces $W^{s,p}$ with $sp\in\mathbb{N}$ and $sp\ge 2$. A sympathetic reader would care because this closes the weak-approximation problem for integer exponents: density of smooth maps in the weak topology can no longer be expected once the relevant homotopy condition fails, even on a ball.

What carries the argument

The central object is the periodic singular retraction $u:\mathbb{R}^{p+1}\setminus\Sigma\to\tilde N_0$ into the $p$-skeleton of the standard cubical decomposition, defined on each unit cube centered at $\sigma\in\Sigma$ by $u(x)=\sigma+(x-\sigma)/(2|x-\sigma|_\infty)$, where $\Sigma=(\mathbb{Z}+1/2)^{p+1}$. The paper's proof that the relaxed energy cannot match the Sobolev energy rests on two structural ingredients. The first is a finite-scale bubbling statement (Proposition 2.1): two continuous Sobolev maps with bounded energy that are close in average distance are homotopic outside finitely many small balls, and the energy inside those balls controls the homotopy gap; this localizes the difference between an approximating sequence and the singular limit. The second is a conical joint estimate on Brouwer degrees (Proposition 3.2), which bounds the total degree with respect to a lattice of singularities by an energy integral over cones; applied to the boundary of cubes in Proposition 3.5, it yields the defect lower bound (3.14) involving the difference $|Dv|^p-|D(\Theta_{\ell,\alpha}\circ v)|^p$, where $\Theta_{\ell,\alpha}$ is the retraction onto the cube $Q_{\ell,\alpha}$. Iterating the inequality $E^{\mathrm{rel}}(Q_{5\ell})\ge 5^{p+1}E^{\mathrm{rel}}(Q_\ell)+c\ell^{p+1}$ produces the log-factor growth. For the sphere target, the analogous mechanism is the Whitehead product construction of a periodic map with Hopf invariant $2$ per singularity, controlled by an integral estimate on the Hopf degree.

What would settle it

Compute or bound from above the relaxed energy of the periodic singular retraction for the smallest case $p=2$, $n=3$: if one can exhibit, for every $\ell$, a sequence of smooth maps $u_k$ into $N_0$ (or into the compact manifold $N_\lambda$) with $E(u_k,Q_\ell)\le C\ell^3$ and $u_k\to u$ a.e., then $E^{\mathrm{rel}}(u,Q_\ell)$ would be $O(\ell^3)$ and the claimed $\ell^3\ln\ell$ growth\u2014and with it the main theorem\u2014would be refuted. Equivalently, produce two maps $u,v$ satisfying the hypotheses of Proposition 2.1 for which no finite family of disjoint balls with the stated energy and homotopy properties exists; that would break the load-bearing bubbling premise.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that analytical obstructions to weak density of smooth maps are a general phenomenon, occurring for every integer exponent $p\ge 2$. The proof identifies the mechanism: for the singular retraction $u$ of $\mathbb{R}^{p+1}$ onto the $p$-skeleton $\tilde N_0$ of the standard cubical lattice, the relaxed energy satisfies $\liminf_{\ell\to\infty} E^{1,p}_{\mathrm{rel}}(u,Q_\ell)/\ell^{p+1}=\infty$, although the Dirichlet energy $E^{1,p}(u,Q_\ell)$ is comparable to $\ell^{p+1}$. The superlinear growth is obtained by iterating a strengthened self-similarity estimate $E^{1,p}_{\mathrm{rel}}(u,Q_{5\ell})\ge 5^{p+1}E^{1,p}_{\mathrm{rel}}(u,Q_\ell)+c\ell^{p+1}$, whose proof uses a finite-scale bubbling proposition for Sobolev maps into uniform Lipschitz neighborhood retracts and a conical joint estimate on Brouwer degrees. The relaxed-energy gap is transferred from the noncompact skeleton to a compact manifold $N_\lambda$ (a level set of an explicit function on $T^{p+1}\times\mathbb{R}^m$) and then, via a nonlinear uniform boundedness principle, to the existence of a map in $W^{1,p}(M,N)$ that no sequence of smooth maps can weakly approximate. In the sphere case, the singular map is built by a Whitehead product with Hopf invariant $2$ for each lattice singularity, and an integral estimate on the Hopf invariant supplies the corresponding degree control on bubbles.

Load-bearing premise

The whole argument leans on a finite-scale bubbling statement: if two Sobolev maps into the target have bounded energy and are very close on average, then away from finitely many tiny balls they are homotopic, with the homotopy gap charged by the energy inside those balls; if that statement fails at the scales chosen in the proof, the extra log of relaxed energy disappears and the obstruction would not follow.

Editorial extensions

If this is right

  • For every integer $p\ge 2$ there is a compact target manifold $N$ (e.g. $T^{p+1}\#T^{p+1}$) such that weak approximation by smooth maps fails as soon as $\dim M>p$; this shows that the positive results known under the hypothesis $\pi_1(N)=\cdots=\pi_{p-1}(N)=0$ cannot be extended to arbitrary targets.
  • When $p=4n-1$, the target can be taken to be the sphere $S^{2n}$; this gives an infinite family of counterexamples, including the earlier $S^2$ case, and shows that the earlier restriction to $p=3$ was an artifact of the available Hopf-invariant-one maps, since the new construction uses Hopf invariant $2$.
  • The obstruction already appears for continuous Sobolev maps, so the failure is not due to a lack of regularity of approximating sequences; the non-weakly-approximable map can be chosen with values in a compact manifold, not merely in a cell complex or a noncompact skeleton.
  • The same conclusion holds in higher-order Sobolev spaces $W^{s,p}$ whenever $sp\in\mathbb{N}$ and $sp\ge 2$ with $\dim M>sp$, so the analytical obstruction persists across all Sobolev regularities for which the question is open.
  • The proof shows why a natural strategy for weak density\u2014passing to the universal cover to trivialize $\pi_1$\u2014cannot work in general: the homotopy between an approximant and the singular limit would require eliminating arbitrarily large cycles in the cover, and the quantitative control needed for that elimination is exactly what the bubbling analysis rules out.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The $\ell^{p+1}\ln\ell$ growth of the relaxed energy suggests a logarithmic \u201centropy cost\u201d for smoothing a periodic lattice of singularities: the cost is proportional to the boundary area of the domain times the logarithm of the system size. If the mechanism is robust, for $p=2$ one would expect the minimal energy of smooth approximants on a cube of side $\ell$ to exceed the singular map
  • Because the constructed target retracts onto the $p$-skeleton of a $(p+1)$-torus, the same mechanism may appear for any target whose universal cover has a $p$-skeleton that is not uniformly retractible; nilmanifolds with torsion-free nilpotent fundamental group are natural next candidates to check.
  • The finite-scale bubbling proposition is stated for uniform Lipschitz neighborhood retracts, so it may have independent uses in quantitative homotopy problems for variational functionals, for instance in estimating minimal connection energies in Ginzburg\u2013Landau type models, provided the constants can be made effective.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proves that for every integer p≥2 there exists a compact Riemannian manifold N such that for every M with dim M>p, the sequential weak closure of smooth maps in W^{1,p}(M,N) is a proper subset of W^{1,p}(M,N). The proof is quantitative: it constructs a periodic singular retraction onto the p-skeleton of the (p+1)-torus and shows its relaxed energy on cubes of side ℓ grows like ℓ^{p+1} ln ℓ while its Sobolev energy grows like ℓ^{p+1}, then applies a nonlinear uniform boundedness principle. For p=4n−1 the target can be S^{2n}, via a Whitehead-product construction with Hopf invariant 2. The paper also states extensions to higher-order spaces W^{s,p} with sp∈N\{0,1} (Theorems 1.3 and 1.4) and provides a proof sketch in Section 5.

Significance. The first-order results are a major advance: they settle the weak approximation problem in the negative for every integer exponent p≥2, generalizing Bethuel's p=3 counterexample and showing that analytical obstructions are not a sporadic phenomenon. The detailed estimates for the relaxed energy, the bubbling proposition (Proposition 2.1), and the use of a uniform boundedness principle are valuable tools. The paper is carefully written and the proofs of Theorems 1.1 and 1.2 appear sound. However, the advertised higher-order results are not proved at the same level: Theorem 1.4 is explicitly not proved, and Theorem 1.3 is only sketched. The paper as submitted therefore substantiates the first-order claims but overclaims the full scope.

major comments (3)
  1. [Section 5, Theorem 1.4] Theorem 1.4 is stated in the abstract and in the introduction as one of the paper's main outcomes, but its proof is reduced to the sentence “we omit the details.” This is not an automatic consequence of the first-order sphere construction: it requires a smooth periodic approximant of the Whitehead product map with derivative bounds near the singularity set (an analogue of Lemma 5.1 for the sphere construction), and it requires the Hopf-degree transport statement of Proposition 4.2 and Corollary 4.3 to remain valid under the Gagliardo–Nirenberg energy comparison (5.2). Neither step is written. Because the theorem is advertised as proved, the manuscript as submitted overclaims its scope.
  2. [Section 5, proof of Theorem 1.3] The proof of Theorem 1.3 is a sketch rather than a complete proof. In particular, the claim that Proposition 3.5 “remains valid” when condition (3.13) is weakened to u(Q_{ℓ,α})⊂Q_{ℓ,α}+B_{C_1 ε} is not demonstrated, and this weakened lower bound is exactly what transfers the relaxed-energy growth from v to v_lip. While the statement is plausible for sufficiently small ε, the verification should be supplied if Theorem 1.3 is to be counted among the paper's theorems; otherwise the result should be presented as conditional.
  3. [Section 3.5, Remark 3.10] The topological description N_λ ≃ (T^n\ B^n)×S^{m−1} ∪_∂ S^{n−1}×B^m and the connected-sum description for m=1 are introduced with the phrase “without giving detailed arguments.” These descriptions are not needed for the proof of the counterexample, but they are used in the introduction to advertise the structure of the target manifold; the authors should either provide the missing details or clearly label these assertions as heuristic.
minor comments (5)
  1. [Section 1.1] There is a typo: “strong appproximation” should be “strong approximation.”
  2. [Section 4.1] In the proof of Proposition 4.1, “Whithead product” should be “Whitehead product.”
  3. [Section 3.5, definition of V] The formula for V(x) should be parenthesized, e.g. V(x):=∑_{j=1}^n (1+x_{2j-1})/2 + ∑_{j=1}^m |x_{2n+j}|^2, to avoid ambiguity in the scope of the product.
  4. [Section 3.5, Remark 3.10] The notation B_{√λ/2}(0) should specify that this is the ball in the R^m factor of T^n × R^m, since the ambient Euclidean space has dimension 2n+m.
  5. [Section 5] The sentence “the case s∉N might be somehow more subtle” is followed by a brief additivity argument; since the paper otherwise gives precise estimates, a slightly more explicit treatment of the fractional Gagliardo–Nirenberg step would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the relaxed-energy counterexample is derived from first-principles degree estimates and external uniform boundedness principles; self-citations are background or supporting theorems, not load-bearing restatements.

full rationale

The proof chain is self-contained against external mathematical benchmarks. The main obstruction (Theorem 1.1) rests on Proposition 3.6, which derives the superlinear growth of the relaxed energy from Proposition 3.5; Proposition 3.5 combines the bubbling Proposition 2.1, proved in the paper from the Morrey–Sobolev embedding and a VMO homotopy criterion, with the conical joint degree estimate of Proposition 3.2, proved by a direct integral estimate. No fitted parameter is renamed as a prediction, and no step assumes the target inclusion H^{1,p}_W(M,N) ⊊ W^{1,p}(M,N). The uniform boundedness principle invoked at the end ([33, Th. 9.6] and [47]) is an external theorem whose hypotheses concern energy scaling and boundedness, not the target conclusion; the self-cited [47] is therefore supporting, not load-bearing circularity. The Whitehead-product construction for Theorem 1.2 uses classical Hopf-invariant identities and Rivière's estimate, again external. The paper does contain an explicitly flagged omitted proof: in Section 5, for Theorem 1.4 it states 'Since the procedure is actually simpler than for Theorem 1.1... we omit the details.' This is a completeness gap in the manuscript, not a circular step, because no reduction to prior assumptions or fitted data is involved. Accordingly, no circularity is found.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper makes no empirical fits and introduces no ad hoc physical entities. It relies on standard background theorems in Sobolev space theory, algebraic topology, and calculus of variations, all cited explicitly. The construction parameters m and λ are free choices in the target manifold, but they are not tuned to data and do not affect the truth of the existence result.

assumptions (6)
  • standard math Nash isometric embedding theorem
    Used in Section 1.1 to embed the target manifold N into Euclidean space in the definition of W^{1,p}(M,N).
  • standard math Nonlinear uniform boundedness principle for weak approximation
    Invoked at the end of the proofs of Theorems 1.1 and 1.2 (via [33, Theorem 9.6] and [47]) to convert superlinear relaxed energy growth into the existence of a non-approximable map.
  • standard math Rivière's Hopf invariant estimate
    Used in Proposition 4.2 to bound the partial Hopf degrees on faces of the (4n-1)-skeleton by Sobolev energy.
  • standard math Branched optimal transport lower bound
    Used in Corollary 4.3 to derive the ℓ^{4n} ln ℓ lower bound from the face degree estimates.
  • standard math Uniqueness of liftings in Sobolev covering spaces
    Used in Proposition 3.8 to identify the weak limit of lifted approximating maps with u+a, a deck transformation.
  • standard math Gagliardo-Nirenberg interpolation inequality
    Used in Section 5 to reduce W^{s,p} energy estimates to W^{1,sp} estimates for the higher-order theorems.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Analytical obstructions to the weak approximation of Sobolev mappings into manifolds." pith.science (2026). https://pith.science/paper/JMBGZS33

@misc{pith2026241212889,
  author       = {Pith},
  title        = {Pith review of: Analytical obstructions to the weak approximation of Sobolev mappings into manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JMBGZS33}},
  note         = {Machine review of arXiv:2412.12889}
}
abstract

For any integer $ p \geq 2 $, we construct a compact Riemannian manifold $ \mathcal{N} $ such that if $ \dim \mathcal{M} > p $, there is a map in the Sobolev space of mappings $ W^{1,p} (\mathcal{M}, \mathcal{N})$ which is not a weak limit of smooth maps into $ \mathcal{N} $ due to a mechanism of analytical obstruction. For $ p = 4n - 1 $, the target manifold can be taken to be the sphere $ \mathbb{S}^{2n} $ thanks to the construction by Whitehead product of maps with nontrivial Hopf invariant, generalizing the result by Bethuel for $ p = 4n -1 = 3$. The results extend to higher order Sobolev spaces $ W^{s,p} $, with $ s \in \mathbb{R} $, $s \geq 1 $, $ sp \in \mathbb{N}$, and $ sp \ge 2 $.

Figures

Figures reproduced from arXiv: 2412.12889 by the authors.

Figure 1
Figure 1. On a generic square, a smooth map uk approximating u should take at most points a value close to u while engulfing at the other points all the singularities through the creation of bubbles. we are going to strengthen this inequality. For this purpose, we consider a sequence (uk)k∈N in W1,p([0, 5ℓ] p+1 , Nf0) ∩ C([0, 5ℓ] p+1 , Nf0) realizing the infimum in (1.6). By a classical Fatou and Fubini–Tonelli argument, the … view at source ↗
Figure 2
Figure 2. Given u and v, the map w is constructed so that it coincides with v outside the larger balls, it is constant on the intermediate sphere and it is a rescaling of v on the smaller balls; the map w0 defined similarly outside the intermediate balls and constant inside those is homotopic to u. (iv) w|∂Bρj/2(aj ) = bj , (v) w(Bρj (aj ) \ Bρj/4 (aj )) ⊆ Bε(bj ), (vi) w = v on M \ SJ j=1 Bρj (aj ), (vii) X J j=1 ˆ Bρj (aj )… view at source ↗
Figure 3
Figure 3. Thanks to an averaging argument, the spheres ∂Bρj (aj ) (in blue) can be chosen out of the growing balls generated by the balls Bρ(a) (in red) so that u and v have a small Sobolev energy and are at small average distance on them. We again assume that the injectivity radius of M is 2ρ∗ with ρ∗ ∈ (0, ∞), which implies a uniform control on the exponential map on any ball Bρ∗ (a), and we assume that ρ0 ≤ ρ∗. Given η ∈ (… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The sets Qℓ,0, Gℓ,□, and Gℓ,γ for γ = (−1, −1). The colored cubes on the boundary form the set Gℓ,□, and the cubes that are further colored in blue form the set Gℓ,γ for γ = (−1, −1). For every α ∈ A := {−2, −1, 0, 1, 2} n , we set Qℓ,α := Qℓ + ℓα + (2ℓ, . . . , 2ℓ), s…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Generic topological screening and approximation of Sobolev maps

    math.FA 2025-01 accept novelty 7.0 of 10

    A Sobolev map between manifolds with kp<m can be approximated by smooth maps exactly when its generic restrictions to floor(kp)-dimensional spheres are VMO-homotopically trivial, and relaxing the target dimension give...

Reference graph

Works this paper leans on

64 extracted references · 40 canonical work pages · cited by 1 Pith paper

  1. [1]

    J. F. Adams,On the non-existence of elements of Hopf invariant one, Ann. of Math. (2)72 (1960), 20–104, doi:10.2307/1970147

    work page doi:10.2307/1970147 1960

  2. [2]

    Alberti, S

    G. Alberti, S. Baldo, and G. Orlandi,Functions with prescribed singularities, J. Eur. Math. Soc. (JEMS) 5 (2003), no. 3, 275–311, doi:10.1007/s10097-003-0053-5

    work page doi:10.1007/s10097-003-0053-5 2003

  3. [3]

    Almgren, W

    F. Almgren, W. Browder, and E. H. Lieb,Co-area, liquid crystals, and minimal surfaces, Partial differential equations, 1988, pp. 1–22, doi:10.1007/BFb0082921

    work page doi:10.1007/bfb0082921 1988

  4. [4]

    Bernot, V

    M. Bernot, V. Caselles, and J.-M. Morel,Optimal transportation networks. Models and theory, Lect. Notes Math., Springer, 2009, doi:10.1007/978-3-540-69315-4

    work page doi:10.1007/978-3-540-69315-4 2009

  5. [5]

    Bethuel, A characterization of maps in H 1(B3,S 2) which can be approximated by smooth maps, Ann

    F. Bethuel, A characterization of maps in H 1(B3,S 2) which can be approximated by smooth maps, Ann. Inst. H. Poincaré C Anal. Non Linéaire7 (1990), no. 4, 269–286, doi:10.1016/S0294- 1449(16)30292-X

    doi:10.1016/s0294- 1990

  6. [6]

    Bethuel,The approximation problem for Sobolev maps between two manifolds, Acta Math.167 (1991), no

    F. Bethuel,The approximation problem for Sobolev maps between two manifolds, Acta Math.167 (1991), no. 3-4, 153–206, doi:10.1007/BF02392449

    work page doi:10.1007/bf02392449 1991

  7. [7]

    Fixed Point Theory Appl.15 (2014), no

    , A new obstruction to the extension problem for Sobolev maps between manifolds, J. Fixed Point Theory Appl.15 (2014), no. 1, 155–183, doi:10.1007/s11784-014-0185-0

    work page doi:10.1007/s11784-014-0185-0 2014

  8. [8]

    Math.219 (2020), no

    , A counterexample to the weak density of smooth maps between manifolds in Sobolev spaces, Invent. Math.219 (2020), no. 2, 507–651, doi:10.1007/s00222-019-00911-3

    work page doi:10.1007/s00222-019-00911-3 2020

Show all 64 references
  1. [9]

    Bethuel and D

    F. Bethuel and D. Chiron,Some questions related to the lifting problem in Sobolev spaces, Perspectives in nonlinear partial differential equations, Contemp. Math., vol. 446, Amer. Math. Soc., Providence, R.I., 2007, pp. 125–152, doi:10.1090/conm/446/08628

    2007 doi

  2. [10]

    Bethuel and F

    F. Bethuel and F. Demengel,Extensions for Sobolev mappings between manifolds, Calc. Var. Partial Differential Equations 3 (1995), no. 4, 475–491, doi:10.1007/BF01187897

    1995 doi

  3. [11]

    Bethuel and X

    F. Bethuel and X. M. Zheng,Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal.80 (1988), no. 1, 60–75, doi:10.1016/0022-1236(88)90065-1

    1988 doi

  4. [12]

    Bourgain, H

    J. Bourgain, H. Brezis, and P. Mironescu,Lifting in Sobolev spaces, J. Anal. Math.80 (2000), 37–86, doi:10.1007/BF02791533. ANALYTICAL OBSTRUCTIONS TO THE WEAK APPROXIMATION OF SOBOLEV MAPPINGS 42

    2000 doi

  5. [13]

    Bousquet, A

    P. Bousquet, A. C. Ponce, and J. Van Schaftingen,Density of smooth maps for fractional Sobolev spacesWs,p into ℓ simply connected manifolds whens≥ 1, Confluentes Math.5 (2013), no. 2, 3–22, doi:10.5802/cml.5

    2013 doi

  6. [14]

    , Strong density for higher order Sobolev spaces into compact manifolds, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 4, 763–817, doi:10.4171/jems/518

    2015 doi

  7. [15]

    , Weak approximation by bounded Sobolev maps with values into complete manifolds, C. R. Math. Acad. Sci. Paris356 (2018), no. 3, 264–271, doi:10.1016/j.crma.2018.01.017

    2018 doi

  8. [16]

    Brezis and J.-M

    H. Brezis and J.-M. Coron,Large solutions for harmonic maps in two dimensions, Comm. Math. Phys. 92 (1983), no. 2, 203–215, doi:10.1007/bf01210846

    1983 doi

  9. [17]

    Brezis, J.-M

    H. Brezis, J.-M. Coron, and E. H. Lieb,Harmonic maps with defects, Comm. Math. Phys.107 (1986), no. 4, 649–705, doi:10.1007/bf01205490

    1986 doi

  10. [18]

    Brezis and P

    H. Brezis and P. Mironescu,Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ.1 (2001), no. 4, 387–404, doi:10.1007/pl00001378

    2001 doi

  11. [19]

    , Density in Ws,p(Ω;N), J. Funct. Anal. 269 (2015), no. 7, 2045–2109, doi:10.1016/j.jfa.2015.04.005

    2015 doi

  12. [20]

    Brezis and L

    H. Brezis and L. Nirenberg,Degree theory and BMO. I: Compact manifolds without boundaries, Selecta Math. (N.S.)1 (1995), no. 2, 197–263, doi:10.1007/BF01671566

    1995 doi

  13. [21]

    Canevari and G

    G. Canevari and G. Orlandi,Topological singular set of vector-valued maps, I: applications to manifold-constrained Sobolev and BV spaces, Calc. Var. Partial Differential Equations58 (2019), no. 2, Paper No. 72, 40, doi:10.1007/s00526-019-1501-8

    2019 doi

  14. [22]

    Da Lio and T

    F. Da Lio and T. Rivière,Three-term commutator estimates and the regularity of1 2-harmonic maps into spheres, Anal. PDE4 (2011), no. 1, 149–190, doi:10.2140/apde.2011.4.149

    2011 doi

  15. [23]

    Deny and J.-L

    J. Deny and J.-L. Lions,Les espaces du type de Beppo Levi, Ann. Inst. Fourier5 (1954), 305–370, doi:10.5802/aif.55

    1954 doi

  16. [24]

    Detaille, A complete answer to the strong density problem in Sobolev spaces with values into compact manifolds(2023), available athttps://arxiv.org/abs/2305.12589

    A. Detaille, A complete answer to the strong density problem in Sobolev spaces with values into compact manifolds(2023), available athttps://arxiv.org/abs/2305.12589

    2023

  17. [25]

    Eells and L

    J. Eells and L. Lemaire,A report on harmonic maps, Bull. London Math. Soc.10 (1978), no. 1, 1–68, doi:10.1112/blms/10.1.1

    1978 doi

  18. [26]

    Eells and J

    J. Eells and J. H. Sampson, Variational theory in fibre bundles, Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965), Nippon Hyoronsha Co., Ltd., Tokyo, 1966, pp. 22–33

    1965

  19. [27]

    J. L. Ericksen and C. Truesdell,Exact theory of stress and strain in rods and shells, Arch. Ration. Mech. Anal. 1 (1958), 295–323, doi:10.1007/BF00298012

    1958 doi

  20. [28]

    Gastel,Partial regularity of polyharmonic maps to targets of sufficiently simple topology, Z

    A. Gastel,Partial regularity of polyharmonic maps to targets of sufficiently simple topology, Z. Anal. Anwend. 35 (2016), no. 4, 397–410, doi:10.4171/ZAA/1571

    2016 doi

  21. [29]

    Giaquinta, G

    M. Giaquinta, G. Modica, and J. Souček, Cartesian currents in the calculus of variations II: Variational integrals, Ergeb. Math. Grenzgeb., Springer, 1998, doi:10.1007/978-3-662-06218-0

    1998 doi

  22. [30]

    Hajłasz, Approximation of Sobolev mappings, Nonlinear Anal

    P. Hajłasz, Approximation of Sobolev mappings, Nonlinear Anal. 22 (1994), no. 12, 1579–1591, doi:10.1016/0362-546X(94)90190-2

    1994 doi

  23. [31]

    Hang,Density problems forW 1,1(M,N ), Comm

    F. Hang,Density problems forW 1,1(M,N ), Comm. Pure Appl. Math.55 (2002), no. 7, 937–947, doi:10.1002/cpa.3020

    2002 doi

  24. [32]

    Hang and F

    F. Hang and F. Lin,Topology of Sobolev mappings. II, Acta Math. 191 (2003), no. 1, 55–107, doi:10.1007/BF02392696

    2003 doi

  25. [33]

    III, Comm

    , Topology of Sobolev mappings. III, Comm. Pure Appl. Math.56 (2003), no. 10, 1383–1415, doi:10.1002/cpa.10098

    2003 doi

  26. [34]

    Hardt and T

    R. Hardt and T. Rivière,Connecting topological Hopf singularities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 2, 287–344

    2003

  27. [35]

    200 (2008), no

    , Connecting rational homotopy type singularities, Acta Math. 200 (2008), no. 1, 15–83, doi:10.1007/s11511-008-0023-6

    2008 doi

  28. [36]

    Hardt and T

    R. Hardt and T. Rivière,Sequential weak approximation for maps of finite Hessian energy, Calc. Var. Partial Differential Equations54 (2015), no. 3, 2713–2749, doi:10.1007/s00526-015-0881-7

    2015 doi

  29. [37]

    Hatcher,Algebraic topology, Cambridge University Press, Cambridge, 2002

    A. Hatcher,Algebraic topology, Cambridge University Press, Cambridge, 2002

    2002

  30. [38]

    Hélein and J

    F. Hélein and J. C. Wood,Harmonic maps, Handbook of global analysis, Elsevier, Amsterdam, 2008, pp. 417–491, 1213, doi:10.1016/B978-044452833-9.50009-7. ANALYTICAL OBSTRUCTIONS TO THE WEAK APPROXIMATION OF SOBOLEV MAPPINGS 43

    2008 doi

  31. [39]

    Huang, Y

    J. Huang, Y. Tong, H. Wei, and H. Bao,Boundary aligned smooth 3D cross-frame field, ACM Transactions on Graphics30 (2011), no. 6, 1–8, doi:10.1145/2070781.2024177

    2011

  32. [40]

    R. L. Jerrard,Lower bounds for generalized Ginzburg–Landau functionals, SIAM J. Math. Anal.30 (1999), no. 4, 721–746, doi:10.1137/S0036141097300581

    1999 doi

  33. [41]

    Jost,Harmonic maps between surfaces, Lecture Notes in Mathematics, vol

    J. Jost,Harmonic maps between surfaces, Lecture Notes in Mathematics, vol. 1062, Springer-Verlag, Berlin, 1984, doi:10.1007/BFb0100160

    1984 doi

  34. [42]

    J. M. Lee,Introduction to smooth manifolds, 2nd ed., Grad. Texts in Math., vol. 218, Springer, 2013, doi:10.1007/978-1-4419-9982-5

    2013 doi

  35. [43]

    N. D. Mermin,The topological theory of defects in ordered media, Rev. Modern Phys.51 (1979), no. 3, 591–648, doi:10.1103/RevModPhys.51.591

    1979 doi

  36. [44]

    N. G. Meyers and J. Serrin, H = W, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 1055–1056, doi:10.1073/pnas.51.6.1055

    1964 doi

  37. [45]

    Mironescu and J

    P. Mironescu and J. Van Schaftingen,Lifting in compact covering spaces for fractional Sobolev mappings, Anal. PDE14 (2021), no. 6, 1851–1871, doi:10.2140/apde.2021.14.1851

    2021 doi

  38. [46]

    , Trace theory for Sobolev mappings into a manifold, Ann. Fac. Sci. Toulouse Math. (6)30 (2021), no. 2, 281–299, doi:10.5802/afst.1675

    2021 doi

  39. [47]

    Monteil and J

    A. Monteil and J. Van Schaftingen, Uniform boundedness principles for Sobolev maps into manifolds, Ann. Inst. H. Poincaré C Anal. Non Linéaire 36 (2019), no. 2, 417–449, doi:10.1016/j.anihpc.2018.06.002

    2019 doi

  40. [48]

    Nakauchi and S

    N. Nakauchi and S. Takakuwa,A remark onp-harmonic maps, Nonlinear Anal.25 (1995), no. 2, 169–185, doi:10.1016/0362-546X(94)00225-7

    1995 doi

  41. [49]

    Nash,C 1 isometric imbeddings, Ann

    J. Nash,C 1 isometric imbeddings, Ann. of Math. (2)60 (1954), 383–396, doi:10.2307/1969840

    1954 doi

  42. [50]

    , The imbedding problem for Riemannian manifolds, Ann. of Math. (2)63 (1956), 20–63, doi:10.2307/1969989

    1956 doi

  43. [51]

    M. R. Pakzad,Weak density of smooth maps inW 1,1(M,N ) for non-abelianπ1(N), Ann. Global Anal. Geom. 23 (2003), no. 1, 1–12, doi:10.1023/a:1021227017504

    2003 doi

  44. [52]

    M. R. Pakzad and T. Rivière,Weak density of smooth maps for the Dirichlet energy between manifolds, Geom. Funct. Anal.13 (2003), no. 1, 223–257, doi:10.1007/s000390300006

    2003 doi

  45. [53]

    Rivière,Minimizing fibrations andp-harmonic maps in homotopy classes fromS3 into S2, Comm

    T. Rivière,Minimizing fibrations andp-harmonic maps in homotopy classes fromS3 into S2, Comm. Anal. Geom. 6 (1998), no. 3, 427–483, doi:10.4310/cag.1998.v6.n3.a2

    1998 doi

  46. [54]

    Runst,Mapping properties of nonlinear operators in spaces of Triebel-Lizorkin and Besov type, Anal

    T. Runst,Mapping properties of nonlinear operators in spaces of Triebel-Lizorkin and Besov type, Anal. Math. 12 (1986), no. 4, 313–346, doi:10.1007/BF01909369

    1986 doi

  47. [55]

    Sacks and K

    J. Sacks and K. Uhlenbeck,The existence of minimal immersions of2-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1–24, doi:10.2307/1971131

    1981 doi

  48. [56]

    Sandier,Lower bounds for the energy of unit vector fields and applications, J

    E. Sandier,Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal.152 (1998), no. 2, 379–403, doi:10.1006/jfan.1997.3170

    1998

  49. [57]

    Sandier and S

    E. Sandier and S. Serfaty,Vortices in the magnetic Ginzburg–Landau model, Progress in Non- linear Differential Equations and their Applications, vol. 70, Birkhäuser, Boston, Mass., 2007, doi:10.1007/978-0-8176-4550-2

    2007 doi

  50. [58]

    Schoen and K

    R. Schoen and K. Uhlenbeck,Boundary regularity and the Dirichlet problem for harmonic maps, J. Differential Geom. 18 (1983), no. 2, 253–268, doi:10.4310/jdg/1214437663

    1983

  51. [59]

    E. H. Spanier,Algebraic topology, McGraw-Hill, New York-Toronto-London, 1966, doi:10.1007/978-1- 4684-9322-1

    1966 doi

  52. [60]

    Schikorra and J

    A. Schikorra and J. Van Schaftingen,An estimate of the Hopf degree of fractional Sobolev mappings, Proc. Amer. Math. Soc.148 (2020), no. 7, 2877–2891, doi:10.1090/proc/15026

    2020 doi

  53. [61]

    Urakawa,Geometry of biharmonic mappings: Differential geometry of variational methods, World Scientific, Hackensack, N.J., 2019, doi:10.1142/10886

    H. Urakawa,Geometry of biharmonic mappings: Differential geometry of variational methods, World Scientific, Hackensack, N.J., 2019, doi:10.1142/10886

    2019 doi

  54. [62]

    White,Homotopy classes in Sobolev spaces and the existence of energy minimizing maps, Acta Math

    B. White,Homotopy classes in Sobolev spaces and the existence of energy minimizing maps, Acta Math. 160 (1988), no. 1-2, 1–17, doi:10.1007/BF02392271

    1988 doi

  55. [63]

    J. H. C. Whitehead,On adding relations to homotopy groups, Ann. of Math. (2)42 (1941), no. 2, 409–428, doi:10.2307/1968907

    1941 doi

  56. [64]

    G. W. Whitehead, Elements of homotopy theory , Grad. Texts in Math., Springer, 1978, doi:10.1007/978-1-4612-6318-0. ANALYTICAL OBSTRUCTIONS TO THE WEAK APPROXIMATION OF SOBOLEV MAPPINGS 44 (A. Detaille) Universite Claude Bernard Lyon 1, CNRS, Centrale Lyon, INSA Lyon, Universi...

    1978 doi

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.