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# 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

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Volume 11, Issue 2

# Sobolev homeomorphisms with gradients of low rank via laminates

Daniel Faraco
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• Other articles by this author:
/ Carlos Mora-Corral
/ Marcos Oliva
• Corresponding author
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• Other articles by this author:
Published Online: 2016-08-30 | DOI: https://doi.org/10.1515/acv-2016-0009

## Abstract

Let $\mathrm{\Omega }\subset {ℝ}^{n}$ be a bounded open set. Given $2\le m\le n$, we construct a convex function $u:\mathrm{\Omega }\to ℝ$ whose gradient $f=\nabla u$ is a Hölder continuous homeomorphism, f is the identity on $\partial \mathrm{\Omega }$, the derivative Df has rank $m-1$ a.e. in Ω and Df is in the weak ${L}^{m}$ space ${L}^{m,w}$. The proof is based on convex integration and staircase laminates.

MSC 2010: 46E35; 25B25; 25B35

## References

• [1]

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

• [2]

K. Astala, D. Faraco and L. Székelyhidi, Jr., Convex integration and the ${L}^{p}$ 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.

• [4]

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

• [5]

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

• [6]

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

• [7]

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

• [8]

S. Conti, D. Faraco and F. Maggi, A new approach to counterexamples to ${L}^{1}$ 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.

• [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.

• [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.

• [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.

• [20]

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

• [21]

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

• [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.

• [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.

• [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.

• [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.

• [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.

• [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

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,

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