Comptes Rendus
Research article - Functional analysis, Geometry and Topology
Obstacles for Sobolev-homeomorphisms with low rank: pointwise a.e. vs distributional Jacobians
Woongbae Park1; Armin Schikorra2
1Department of Mathematics, 803 Hylan building, University of Rochester, Rochester, NY 14627, USA
2Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA
Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1025-1033

We show that for any $k$ and $s > {\frac{k}{k+1}}$ there exist neither $W^{s,\frac{k}{s}}$-Sobolev nor $C^s$-Hölder homeomorphisms from the disk $\overline{\mathbb{B}^n}$ into $\mathbb{R}^N$ whose gradient has rank $< k$ in the distributional sense. This complements known examples of such kind of homeomorphisms whose gradient has rank $<k$ almost everywhere.

Nous montrons que pour tout $k$ et $s > {\frac{k}{k+1}}$, il n’existe pas d’homéomorphismes $W^{s,\frac{k}{s}}$-Sobolev ou $C^s$-Hölder du disque $\overline{\mathbb{B}^n}$ dans $\mathbb{R}^N$ dont le gradient a un rang $< k$ au sens distributionnel. Ceci complète les exemples connus de ce type d’homéomorphismes dont le gradient a un rang $< k$ presque partout.

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DOI: 10.5802/crmath.771
Classification: 46E35, 55M25

Woongbae Park  1 ; Armin Schikorra  2

1 Department of Mathematics, 803 Hylan building, University of Rochester, Rochester, NY 14627, USA
2 Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Woongbae Park; Armin Schikorra. Obstacles for Sobolev-homeomorphisms with low rank: pointwise a.e. vs distributional Jacobians. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1025-1033. doi: 10.5802/crmath.771

[1] John M. Ball Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., Volume 63 (1976/77) no. 4, pp. 337-403 | DOI | MR

[2] Haïm Brezis; Hoai-Minh Nguyen The Jacobian determinant revisited, Invent. Math., Volume 185 (2011) no. 1, pp. 17-54 | DOI | MR | Zbl

[3] Haïm Brezis; L. Nirenberg Degree theory and BMO. I. Compact manifolds without boundaries, Sel. Math., New Ser., Volume 1 (1995) no. 2, pp. 197-263 | DOI | MR | Zbl

[4] R. Coifman; P.-L. Lions; Y. Meyer; S. Semmes Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9), Volume 72 (1993) no. 3, pp. 247-286 | MR | Zbl

[5] Daniel Faraco; Carlos Mora-Corral; Marcos Oliva Sobolev homeomorphisms with gradients of low rank via laminates, Adv. Calc. Var., Volume 11 (2018) no. 2, pp. 111-138 | DOI | MR | Zbl

[6] Peter Gladbach; Heiner Olbermann Coarea formulae and chain rules for the Jacobian determinant in fractional Sobolev spaces, J. Funct. Anal., Volume 278 (2020) no. 2, 108312, 21 pages | DOI | MR | Zbl

[7] Mikhael Gromov Carnot–Carathéodory spaces seen from within, Sub-Riemannian geometry (Progress in Mathematics), Birkhäuser, 1996 no. 144, pp. 79-323 | Zbl | MR | DOI

[8] Piotr Hajłasz Linking topological spheres, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl., Volume 30 (2019) no. 4, pp. 907-909 | Zbl | DOI | MR

[9] Piotr Hajłasz; Jacob Mirra; Armin Schikorra Hölder continuous mappings, differential forms and the Heisenberg groups (2025) | arXiv | Zbl

[10] Piotr Hajłasz; Armin Schikorra On the Gromov non-embedding theorem (2023) | arXiv | Zbl

[11] Enno Lenzmann; Armin Schikorra Sharp commutator estimates via harmonic extensions, Nonlinear Anal., Theory Methods Appl., Volume 193 (2020), 111375, 37 pages | DOI | MR | Zbl

[12] Giovanni Leoni A first course in fractional Sobolev spaces, Graduate Studies in Mathematics, 229, American Mathematical Society, 2023, xv+586 pages | DOI | MR | Zbl

[13] Siran Li; Armin Schikorra W s,n s -maps with positive distributional Jacobians, Potential Anal., Volume 55 (2021) no. 3, pp. 403-417 | DOI | MR | Zbl

[14] Zhuomin Liu; Jan Malý A strictly convex Sobolev function with null Hessian minors, Calc. Var. Partial Differ. Equ., Volume 55 (2016) no. 3, 58, 19 pages | DOI | MR | Zbl

[15] Alexander Lytchak; Stefan Wenger; Robert Young Dehn functions and Hölder extensions in asymptotic cones, J. Reine Angew. Math., Volume 763 (2020), pp. 79-109 | DOI | MR | Zbl

[16] Stefan Müller Higher integrability of determinants and weak convergence in L 1 , J. Reine Angew. Math., Volume 412 (1990), pp. 20-34 | DOI | MR | Zbl

[17] Ju. G. Rešetnjak The weak convergence of completely additive vector-valued set functions, Sib. Mat. Zh., Volume 9 (1968), pp. 1386-1394 | MR | Zbl

[18] Armin Schikorra Hölder-topology of the Heisenberg group, Aequationes Math., Volume 94 (2020) no. 2, pp. 323-343 | DOI | MR | Zbl

[19] Armin Schikorra; Jean Van Schaftingen An estimate of the Hopf degree of fractional Sobolev mappings, Proc. Am. Math. Soc., Volume 148 (2020) no. 7, pp. 2877-2891 | DOI | MR | Zbl

[20] Winfried Sickel; Abdellah Youssfi The characterisation of the regularity of the Jacobian determinant in the framework of potential spaces, J. Lond. Math. Soc. (2), Volume 59 (1999) no. 1, pp. 287-310 | DOI | MR | Zbl

[21] L. Tartar Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot–Watt Symposium, Vol. IV (Research Notes in Mathematics), Pitman Advanced Publishing Program, 1979 no. 39, pp. 136-212 | MR | Zbl

[22] J. W. Vick Homology theory, Graduate Texts in Mathematics, 145, Springer, 1994, xiv+242 pages | DOI | MR | Zbl

[23] Stefan Wenger; Robert Young Constructing Hölder maps to Carnot groups, J. Am. Math. Soc., Volume 38 (2025) no. 2, pp. 291-318 | DOI | MR | Zbl

[24] Henry C. Wente An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl., Volume 26 (1969), pp. 318-344 | DOI | MR | Zbl

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