Rigidity of Euclidean product structure: breakdown for low Sobolev exponents

Bruce Kleiner; Stefan Müller; László Székelyhidi Jr.; Xiangdong Xie

doi:10.3934/cpaa.2024029

Article Contents

2024, Volume 23, Issue 10: 1569-1607. Doi: 10.3934/cpaa.2024029

This issue Previous Article Remark on the stability of energy maximizers for the 2D Euler equation on $ \mathbb{T}^2 $ Next Article Lower semicontinuity, Stoilow factorization and principal maps

Rigidity of Euclidean product structure: breakdown for low Sobolev exponents

1.
Courant Institute of Mathematical Sciences, New York University, USA
2.
Institute for Applied Mathematics, University of Bonn, Germany
3.
Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
4.
Department of Mathematics and Statistics, Bowling Green State University, USA

^*Corresponding author: Stefan Müller
^*Corresponding author: Stefan Müller

Dedicated to Professor Vladimir Šverák on the occasion of his 65th birthday

Received: August 2023

Revised: February 2024

Early access: April 19, 2024

Published online: April 19, 2024

BK was supported by NSF grants DMS-1711556 and DMS-2005553, and a Simons Collaboration grant.
Stefan Müller has been supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the Hausdorff Center for Mathematics (GZ EXC 59 and 2047/1, Projekt-ID 390685813) and the collaborative research center The mathematics of emerging effects (CRC 1060, Projekt-ID 211504053).
László Székelyhidi Jr. gratefully acknowledges the support of Grant Agreement No. 724298-DIFFINCL of the European Research Council and the support of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through GZ SZ 325/2-1.
Xiangdong Xie has been supported by Simons Foundation grant # 315130.

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Abstract

The purpose of this paper is twofold. First, we show that the results in a companion paper on product rigidity for maps $ f: \Omega_1 \times \Omega_2 \subset \mathbb R^n \times \mathbb R^n \to \mathbb R^n \times \mathbb R^n $ in the Sobolev space $ W^{1, p} $ are sharp with respect to $ p $. Specifically, we show that for all $ n \ge 2 $ and all $ p < 2 $ there exist maps $ f \in W^{1, p} $ such that the weak differential $ Df $ is invertible almost everywhere and preserves or reverses the product structure almost everywhere, but $ f $ is not of the form $ f(x, y) = (f_1(x), f_2(y)) $ or = $ f(x, y) = (f_2(y), f_1(x)) $. Secondly, we develop a general toolbox to study $ W^{1, p} $ solutions of differential inclusions $ Du \in K $ for unbounded sets $ K $. As an illustration we give short proofs of results by Astala et al. on optimal $ W^{1, p} $ regularity for solutions of elliptic equations with measurable coefficients and results by Colombo and Tione on irregular solutions of the $ p $-Laplace equation.

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Mathematics Subject Classification: Primary: 35R70, 35A35; Secondary: 30C65, 46E35.

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[1]	K. Astala, D. Faraco and Jr. L. Székelyhidi, Convex integration and the $L^p$ theory of elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 7 (2008), 1-50.
[2]	K. Astala, Area distortion of quasiconformal mappings, Acta Math., 173 (1994), 37-60. doi: 10.1007/BF02392568.
[3]	N. Boros, Jr. L. Székelyhidi and A. Volberg, Laminates meet Burkholder functions, J. Math. Pures Appl., 100 (2013), 687-700. doi: 10.1016/j.matpur.2013.01.017.
[4]	A. Cellina, On minima of a functional of the gradient: necessary conditions, Nonlinear Anal., 20 (1993), 337-341. doi: 10.1016/0362-546X(93)90137-H.
[5]	A. Cellina, A view on differential inclusions, Rend. Semin. Mat. Univ. Politec. Torino, 63 (2005), 197-209.
[6]	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), 287-300. doi: 10.1007/s00205-004-0350-5.
[7]	S. Conti, D. Faraco and F. Maggi, et al., Rank-one convex functions on 2 × 2 symmetric matrices and laminates on rank-three lines, Calc. Var. Partial Differ. Equ., 24 (2005), 479-493.
[8]	M. Colombo and R. Tione, Non-classical solutions of the $p$-laplace equation, arXiv: 2201.07484, 2022.
[9]	B. Dacorogna, Remarques sur les notions de polyconvexité, quasi-convexité et convexité de rang 1, J. Math. Pures Appl., 64 (1985), 403-438.
[10]	B. Dacorogna and P. Marcellini, Implicit Partial Differential Equations, Progress in Nonlinear Differential Equations and their Applications, vol. 37, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1562-2.
[11]	B. Dacorogna, P. Marcellini and E. Paolini, Lipschitz-continuous local isometric immersions: rigid maps and origami, J. Math. Pures Appl., 90 (2008), 66-81. doi: 10.1016/j.matpur.2008.02.011.
[12]	D. Faraco, Milton's conjecture on the regularity of solutions to isotropic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 889-909. doi: 10.1016/S0294-1449(03)00014-3.
[13]	D. Faraco, S. Lindberg and L. Székelyhidi, Magnetic helicity, weak solutions and relaxation of ideal mhd, Commun. Pure Appl. Math., 77 (2024), 2387-2412. doi: 10.1002/cpa.22168.
[14]	D. Faraco, C. Mora-Corral and M. Oliva, Sobolev homeomorphisms with gradients of low rank via laminates, Adv. Calc. Var., 11 (2018), 111-138. doi: 10.1515/acv-2016-0009.
[15]	M. Gromov, Partial Differential Relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-662-02267-2.
[16]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition.
[17]	B. Kirchheim, Analysis and Geometry of Microstructure, Habilitation thesis, University of Leipzig, https://www.mis.mpg.de/preprints/ln/lecturenote-1603.pdf, 2003.
[18]	B. Kleiner, S. Müller and Jr. L. Székelyhidi, et al., Sobolev mappings of Euclidean space and product structure, in preparation, 2023.
[19]	B. Kleiner, S. Müller and X. Xie, Pansu pullback and rigidity of mappings between Carnot groups, arXiv: 2004.09271, 2020.
[20]	B. Kirchheim, V. Šverák and S. Müller, Studying nonlinear pde by geometry in matrix space, in Geometric Analysis and Nonlinear Partial Differential Equations, Springer, Berlin, 2003,347-395.
[21]	Z. M. Liu and J. Malý, A strictly convex Sobolev function with null Hessian minors, Calc. Var. Partial Differ. Equ., 55 (2016), 19 pp. doi: 10.1007/s00526-016-0994-7.
[22]	S. Müller and V. Šverák, Attainment results for the two-well problem by convex integration, Geometric analysis and the calculus of variations, Int. Press, Cambridge, MA, 1996.
[23]	S. Müller and M. A. Sychev, Optimal existence theorems for nonhomogeneous differential inclusions, J. Funct. Anal., 181 (2001), 447-475. doi: 10.1006/jfan.2000.3726.
[24]	S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity, Ann. Math., 157 (2003), 715-742. doi: 10.4007/annals.2003.157.715.
[25]	S. Müller, Variational models for microstructure and phase transitions, Calculus of variations and geometric evolution problems (Cetraro, 1996), Lecture Notes in Math., vol. 1713, Springer, Berlin, 1999, 85-210. doi: 10.1007/BFb0092670.
[26]	M. Oliva, Bi-Sobolev homeomorphisms $f$ with $Df$ and $Df^{-1}$ of low rank using laminates, Calc. Var. Partial Differ. Equ., 55 (2016), 38 pp. doi: 10.1007/s00526-016-1080-x.
[27]	P. Pedregal, Laminates and microstructure, European J. Appl. Math., 4 (1993), 121-149. doi: 10.1017/S0956792500001030.
[28]	W. Pompe, Explicit construction of piecewise affine mappings with constraints, Bull. Pol. Acad. Sci. Math., 58 (2010), 209-220. doi: 10.4064/ba58-3-4.
[29]	M. A. Sychev, Comparing two methods of resolving homogeneous differential inclusions, Calc. Var. Partial Differ. Equ., 13 (2001), 213-229. doi: 10.1007/PL00009929.
[30]	Jr. L. Székelyhidi, Counterexamples to elliptic regularity and convex integration, The Interaction of Analysis and Geometry, Contemp. Math., vol. 424, Amer. Math. Soc., Providence, RI, 2007. doi: 10.1090/conm/424/08104.