Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang


IMPACT FACTOR 2018: 2.316

CiteScore 2018: 1.77

SCImago Journal Rank (SJR) 2018: 2.350
Source Normalized Impact per Paper (SNIP) 2018: 1.465

Mathematical Citation Quotient (MCQ) 2018: 1.44

Online
ISSN
1864-8266
See all formats and pricing
More options …
Volume 11, Issue 2

Issues

Sobolev homeomorphisms with gradients of low rank via laminates

Daniel Faraco
  • Department of Mathematics, Faculty of Sciences, Universidad Autónoma de Madrid, E-28049 Madrid; and ICMAT CSIC-UAM-UCM-UC3M, E-28049 Madrid, Spain
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Carlos Mora-Corral / Marcos Oliva
  • Corresponding author
  • Department of Mathematics, Faculty of Sciences, Universidad Autónoma de Madrid, E-28049 Madrid; and ICMAT CSIC-UAM-UCM-UC3M, E-28049 Madrid, Spain
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-08-30 | DOI: https://doi.org/10.1515/acv-2016-0009

Abstract

Let Ωn be a bounded open set. Given 2mn, we construct a convex function u:Ω whose gradient f=u is a Hölder continuous homeomorphism, f is the identity on Ω, the derivative Df has rank m-1 a.e. in Ω and Df is in the weak Lm space Lm,w. The proof is based on convex integration and staircase laminates.

Keywords: Sobolev homeomorphisms; low rank; laminates; Luzin condition

MSC 2010: 46E35; 25B25; 25B35

References

  • [1]

    G. Alberti and L. Ambrosio, A geometrical approach to monotone functions in n, Math. Z. 230 (1999), no. 2, 259–316. Google Scholar

  • [2]

    K. Astala, D. Faraco and L. Székelyhidi, Jr., Convex integration and the Lp theory of elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7 (2008), no. 1, 1–50. Google Scholar

  • [3]

    J. M. Ball, Global invertibility of Sobolev functions and the interpenetration of matter, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), no. 3–4, 315–328. CrossrefGoogle Scholar

  • [4]

    N. Boros, L. Székelyhidi, Jr. and A. Volberg, Laminates meet Burkholder functions, J. Math. Pures Appl. (9) 100 (2013), no. 5, 687–700. CrossrefWeb of ScienceGoogle Scholar

  • [5]

    Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), no. 4, 375–417. CrossrefGoogle Scholar

  • [6]

    R. Černý, Homeomorphism with zero Jacobian: Sharp integrability of the derivative, J. Math. Anal. Appl. 373 (2011), no. 1, 161–174. Web of ScienceCrossrefGoogle Scholar

  • [7]

    R. Černý, Bi-Sobolev homeomorphism with zero minors almost everywhere, Adv. Calc. Var. 8 (2015), no. 1, 1–30. CrossrefWeb of ScienceGoogle Scholar

  • [8]

    S. Conti, D. Faraco and F. Maggi, A new approach to counterexamples to L1 estimates: Korn’s inequality, geometric rigidity, and regularity for gradients of separately convex functions, Arch. Ration. Mech. Anal. 175 (2005), no. 2, 287–300. Google Scholar

  • [9]

    S. Conti, D. Faraco, F. Maggi and S. Müller, Rank-one convex functions on 2×2 symmetric matrices and laminates on rank-three lines, Calc. Var. Partial Differential Equations 24 (2005), no. 4, 479–493. Google Scholar

  • [10]

    B. Dacorogna, Direct Methods in the Calculus of Variations, Appl. Math. Sci. 78, Springer, Berlin, 1989. Google Scholar

  • [11]

    G. De Philippis and A. Figalli, The Monge–Ampère equation and its link to optimal transportation, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 4, 527–580. CrossrefGoogle Scholar

  • [12]

    K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. Google Scholar

  • [13]

    L. D’Onofrio, S. Hencl and R. Schiattarella, Bi-Sobolev homeomorphism with zero Jacobian almost everywhere, Calc. Var. Partial Differential Equations 51 (2014), no. 1–2, 139–170. CrossrefWeb of ScienceGoogle Scholar

  • [14]

    L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, Boca Raton, 1992. Google Scholar

  • [15]

    D. Faraco, Milton’s conjecture on the regularity of solutions to isotropic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), no. 5, 889–909. CrossrefGoogle Scholar

  • [16]

    D. Faraco, Wild mappings built on unbounded laminates, Proceedings of the Workshop “New Developments in the Calculus of Variations”, Sezione Statist. Mat. 2, Edizioni Scientifiche Italiane, Napoli (2006), 89–108. Google Scholar

  • [17]

    H. Federer, Geometric Measure Theory, Grundlehren Math. Wiss. 153, Springer, New York, 1969. Google Scholar

  • [18]

    I. Gohberg, S. Goldberg and M. A. Kaashoek, Classes of Linear Operators. Vol. I, Oper. Theory Adv. Appl. 49, Birkhäuser, Basel, 1990. Google Scholar

  • [19]

    P. Hajłasz, Change of variables formula under minimal assumptions, Colloq. Math. 64 (1993), no. 1, 93–101. CrossrefGoogle Scholar

  • [20]

    S. Hencl, Sobolev homeomorphism with zero Jacobian almost everywhere, J. Math. Pures Appl. (9) 95 (2011), no. 4, 444–458. Web of ScienceCrossrefGoogle Scholar

  • [21]

    J. Kauhanen, P. Koskela and J. Malý, Mappings of finite distortion: Condition N, Michigan Math. J. 49 (2001), no. 1, 169–181. CrossrefGoogle Scholar

  • [22]

    B. Kirchheim, Rigidity and Geometry of Microstructures, Habilitation thesis, University of Leipzig, Leipzig, 2003. Google Scholar

  • [23]

    B. Kirchheim and J. Kristensen, Automatic convexity of rank-1 convex functions, C. R. Math. Acad. Sci. Paris 349 (2011), no. 7–8, 407–409. CrossrefGoogle Scholar

  • [24]

    B. Kirchheim and J. Kristensen, On rank one convex functions that are homogeneous of degree one, Arch. Ration. Mech. Anal. 221 (2016), no. 1, 527–558. Web of ScienceCrossrefGoogle Scholar

  • [25]

    P. Koskela, J. Malý and T. Zürcher, Luzin’s condition (N) and modulus of continuity, Adv. Calc. Var. 8 (2015), no. 2, 155–171. Google Scholar

  • [26]

    Z. Liu and J. Malý, A strictly convex Sobolev function with null Hessian minors, Calc. Var. Partial Differential Equations 55 (2016), Article ID 58. Web of ScienceGoogle Scholar

  • [27]

    M. Marcus and V. J. Mizel, Transformations by functions in Sobolev spaces and lower semicontinuity for parametric variational problems, Bull. Amer. Math. Soc. 79 (1973), 790–795. CrossrefGoogle Scholar

  • [28]

    S. Müller, Variational models for microstructure and phase transitions, Calculus of Variations and Geometric Evolution Problems (Cetraro 1996), Lecture Notes in Math. 1713, Springer, Berlin (1999), 85–210. Google Scholar

  • [29]

    S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity, Ann. of Math. (2) 157 (2003), no. 3, 715–742. CrossrefGoogle Scholar

  • [30]

    P. Pedregal, Laminates and microstructure, European J. Appl. Math. 4 (1993), no. 2, 121–149. Google Scholar

  • [31]

    P. Pedregal, Parametrized Measures and Variational Principles, Progr. Nonlinear Differential Equations Appl. 30, Birkhäuser, Basel, 1997. Google Scholar

  • [32]

    C. Villani, Topics in optimal transportation, Grad. Stud. Math. 58, American Mathematical Society, Providence, 2003. Google Scholar

About the article


Received: 2016-03-02

Revised: 2016-07-07

Accepted: 2016-07-21

Published Online: 2016-08-30

Published in Print: 2018-04-01


Funding Source: Ministerio de Economía y Competitividad

Award identifier / Grant number: MTM2014-57769-C3-1-P

Award identifier / Grant number: RYC-2010-06125

Funding Source: European Research Council

Award identifier / Grant number: 307179

The authors have been supported by Project MTM2014-57769-C3-1-P of the Spanish Ministry of Economy and Competitivity and the ERC Starting grant no. 307179. The second author has also been supported by the “Ramón y Cajal” grant RYC-2010-06125 (Spanish Ministry of Economy and Competitivity).


Citation Information: Advances in Calculus of Variations, Volume 11, Issue 2, Pages 111–138, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2016-0009.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Zhuomin Liu, Jan Malý, and Mohammad Reza Pakzad
Nonlinear Analysis, 2018, Volume 176, Page 209
[2]
Silvio Fanzon and Mariapia Palombaro
Calculus of Variations and Partial Differential Equations, 2017, Volume 56, Number 5
[3]
Marcos Oliva
Calculus of Variations and Partial Differential Equations, 2016, Volume 55, Number 6

Comments (0)

Please log in or register to comment.
Log in