Lower semicontinuity, Stoilow factorization and principal maps

Kari Astala; Daniel Faraco; André Guerra; Aleksis Koski; Jan Kristensen

doi:10.3934/cpaa.2024069

Article Contents

2024, Volume 23, Issue 10: 1608-1645. Doi: 10.3934/cpaa.2024069

This issue Previous Article Rigidity of Euclidean product structure: breakdown for low Sobolev exponents Next Article Preface

Lower semicontinuity, Stoilow factorization and principal maps

1.
Department of Mathematics and Statistics, University of Helsinki, Finland
2.
Departamento de Matematicas, Universidad Autonoma de Madrid, Spain and ICMAT CSIC, Madrid, Spain
3.
Institute for Theoretical Studies ETH-ITS, Zürich, Switzerland
4.
Department of Mathematics and Systems Analysis, Aalto University, Espoo, Finland
5.
Mathematical Institute, University of Oxford, United Kingdom

^*Corresponding author
^*Corresponding author

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

Received: January 2024

Revised: July 2024

Early access: August 30, 2024

Published online: August 30, 2024

D. F, K. A, A. K acknowledge the financial support of QUAMAP, the ERC Advanced Grant 834728, and of the Severo Ochoa Programme CEX2019-000904-S. A. G. was supported by Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zürich Foundation. D. F and A. K were partially supported by CM and UAM, and A.K by Academy of Finland CoE Randomness and Structures, and Academy Fellowship Grant 355840. D. F acknowledges financial support by PI2021-124-195NB-C32.

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

We consider a refinement of the usual quasiconvexity condition of Morrey in two dimensions that allows us to prove lower semicontinuity and existence of minimizers for a class of functionals which are unbounded as the determinant vanishes and are non-polyconvex in general. This notion, that we call principal quasiconvexity, arose from the planar theory of quasiconformal mappings and mappings of finite distortion. We compare it with other quasiconvexity conditions that have appeared in the literature and provide a number of concrete examples of principally quasiconvex functionals that are not polyconvex. We also describe local conditions which combined with quasiconvexity yield principal quasiconvexity. The Stoilow factorization, that in the context of maps of integrable distortion was developed by Iwaniec and Šverák, plays a prominent role in our approach.

Keywords:

Mathematics Subject Classification: Primary: 49J45, 30C70; Secondary: 74B20.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Ration. Mech. Anal., 86 (1984), 125-145. doi: 10.1007/BF00275731.
[2]	J. J. Alibert and G. Bouchitté, Non-uniform integrability and generalized Young measures, J. Convex Anal., 4 (1997), 129-147.
[3]	K. Astala, Area distortion of quasiconformal mappings, Acta Math., 173 (1994), 37-60. doi: 10.1007/BF02392568.
[4]	K. Astala and D. Faraco, Quasiregular mappings and Young measures, Proc. R. Soc. Edinb. A: Math, 132 (2002), 1045-1056.
[5]	K. Astala, D. Faraco and A. Guerra, et al., The local Burkholder Functional, Quasiconvexity and Geometric Function Theory, Preprint 2023, arXiv: 2309.03495
[6]	K. Astala, D. Faraco and A. Guerra, et al., Lower semicontinuity of isotropic functionals, work in progress.
[7]	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., 7, (2008), 1-50.
[8]	K. Astala, T. Iwaniec and G.J. Martin, Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton University Press, 2009.
[9]	K. Astala, T. Iwaniec, G. J. Martin and et al., Extremal mappings of finite distortion, Proc. London Math. Soc., 91 (2005), 655-702. doi: 10.1112/S0024611505015376.
[10]	K. Astala, T. Iwaniec and I. Prause, et al., Burkholder integrals, Morrey's problem and quasiconformal mappings, J. Amer. Math. Soc., 25 (2012), 507-531. doi: 10.1090/S0894-0347-2011-00718-2.
[11]	K. Astala, T. Iwaniec, I. Prause and et al., A hunt for sharp $\mathcal{L}^p$-estimates and rank-one convex variational integrals, Filomat, 29 (2015), 245-261. doi: 10.2298/FIL1502245A.
[12]	K. Astala, T. Iwaniec and I. Prause, et al., Bilipschitz and quasiconformal rotation, stretching and multifractal spectra, Publ. Math. IHÉS, 121 (2015), 113-154.
[13]	A. Baernstein and S. Montgomery-Smith, Some conjectures about integral means of $\partial f$ and $\bar{\partial} f$, Complex analysis and differential equations (Uppsala, 1997), 92-109. Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., 64, Uppsala Univ., Uppsala 1999.
[14]	E. J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control Optim., 22 (1984), 570-598. doi: 10.1137/0322035.
[15]	J. M. Ball, Convexity conditions and existence theorems in non-linear elasticity, Arch. Ration. Mech. Anal., 63 (1977), 337-403. doi: 10.1007/BF00279992.
[16]	J. M. Ball, Constitutive inequalities and existence theorems in nonlinear elastostatics, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, (R.J. Knops, editor) Volume 1, pp. 187–241. Pitman, 1977.
[17]	J. M. Ball, Global invertibility of Sobolev functions and the interpenetration of matter, Proc. R. Soc. Edinb. A: Math, 88 (1981), 315-328.
[18]	J. M. Ball, Some open problems in elasticity, Geometry, mechanics, and dynamics, pages 3–59. Springer, New York, NY, 2002.
[19]	J. M. Ball and F. Murat, $W^{1, p}$-quasiconvexity and variational problems for multiple integrals, J. Funct. Anal., 58 (1984), 225-253. doi: 10.1016/0022-1236(84)90041-7.
[20]	J. M. Ball and R. D. James, A characterization of plane strain, Proc. R. Soc. Lond. A, 432 (1991), 93-99. doi: 10.1098/rspa.1991.0006.
[21]	R. Bañuelos, The foundational inequalities of D. L. Burkholder and some of their ramifications, Illinois J. Math., 54 (2010), 789-868.
[22]	R. Bañuelos and P. Janakiraman, $L^p$-bounds for the Beurling-Ahlfors transform, Trans. Amer. Math. Soc., 360 (2008), 3603-3612. doi: 10.1090/S0002-9947-08-04537-6.
[23]	B. Benešová and M. Kružík, Weak lower semicontinuity of integral functionals and applications, SIAM Rev., 59 (2017), 703-766. doi: 10.1137/16M1060947.
[24]	J. Bonet, A. J. Gil and R. Ortigosa, A computational framework for polyconvex large strain elasticity, Comput. Methods Appl. Mech. Engrg., 283 (2015), 1061-1094.
[25]	A. Borichev, P. Janakiraman and A. Volberg, Subordination by conformal martingales in $L^p$ and zeros of Laguerre polynomials, Duke Math. J., 162 (2013), 889-924.
[26]	D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab., 12 (1984), 647-702.
[27]	D. L. Burkholder, Sharp inequalities for martingales and stochastic integrals, Astérisque, 157–158 (1988), 75-94.
[28]	D. Campbell, Diffeomorphic approximation of planar Sobolev homeomorphisms in Orlicz-Sobolev spaces, J. Funct. Anal., 273 (2017), 125-205.
[29]	D. Campbell, A. Kauranen and E. Radici, Classification of strict limits of planar BV homeomorphisms, J. Funct. Anal., 285 (2023), 30 pp.
[30]	P. Charrier, B. Dacorogna and B. Hanouzet, et al., An existence theorem for slightly compressible material in nonlinear elasticity, SIAM J. Math. Anal., 19 (1988), 70-85. doi: 10.1137/0519005.
[31]	C. Y. Chen and J. Kristensen, On coercive variational integrals, Nonlinear Anal., 153 (2017), 213-229. doi: 10.1016/j.na.2016.09.011.
[32]	P. G. Ciarlet, Mathematical Elasticity: Three-Dimensional Elasticity, Society for Industrial and Applied Mathematics, 2021.
[33]	B. Dacorogna, Direct methods in the calculus of variations, Vol. 78, Springer Science & Business Media, 2007.
[34]	G. De Philippis and A. Pratelli, The closure of planar diffeomorphisms in Sobolev spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 181-224. doi: 10.1016/j.anihpc.2019.08.001.
[35]	D. Faraco, Tartar conjecture and Beltrami operators, Michigan Math. J., 52 (2004), 83-104.
[36]	D. Faraco, C. Mora-Corral and M. De la Oliva, Sobolev homeomorphisms with gradients of low rank via laminates, Adv. Calc. Var., 11 (2018), 111-138.
[37]	D. Faraco and L. Székelyhidi Jr., Tartar's conjecture and localization of the quasiconvex hull in $\mathbb{R}^{2\times 2}$, Acta Math., 200 (2008), 279-305. doi: 10.1007/s11511-008-0028-1.
[38]	P. J. Flory, Thermodynamic relations for high elastic materials, Trans. Faraday Soc., 57 (1961), 829-838. doi: 10.1039/tf9615700829.
[39]	I. Fonseca and J. Malý, Relaxation of multiple integrals below the growth exponent, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 14 (1997), 309-338. doi: 10.1016/s0294-1449(97)80139-4.
[40]	I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in $L^1$, SIAM J. Math. Anal., 23 (1992), 1081-1098. doi: 10.1137/0523060.
[41]	N. Fusco, G. Moscariello and C. Sbordone, The limit of $ \mathrm{W}^{1, 1}$ homeomorphisms with finite distortion, Calc. Var., 33 (2008), 377-390. doi: 10.1007/s00526-008-0169-2.
[42]	M. Giaquinta, G. Modica and J. Souček, Cartesian currents in the calculus of variations II: variational integrals, Vol. 2, Springer Science & Business Media, 1998.
[43]	Y. Grabovsky, From microstructure-independent formulas for composite materials to rank-one convex, non-quasiconvex functions, Arch. Ration. Mech. Anal., 227 (2018), 607-636. doi: 10.1007/s00205-017-1169-1.
[44]	L. Greco, T. Iwaniec and G. Moscariello, Limits of the improved integrability of the volume forms, Indiana Univ. Math. J.. 44 (1995), 305-339.
[45]	A. Guerra, Extremal rank-one convex integrands and a conjecture of Šverák, Calc. Var. Partial Differ. Equ., 58 (2019), 1-19. doi: 10.1007/s00526-019-1646-5.
[46]	A. Guerra and J. Kristensen, Automatic quasiconvexity of homogeneous isotropic rank-one convex integrands, Arch. Ration. Mech. Anal., 245 (2022), 479-500. doi: 10.1007/s00205-022-01792-2.
[47]	S. Hartmann and P. Neff, Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility, Int. J. Solids Struct., 40 (2003), 2767-2791. doi: 10.1016/S0020-7683(03)00086-6.
[48]	D. Henao and C. Mora-Corral, Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity, Arch. Ration. Mech. Anal., 197 (2010), 619-655.
[49]	S. Hencl and P. Koskela, Lectures on mappings of finite distortion, Vol. 2096, Springer, 2014.
[50]	S. Hencl and A. Pratelli, Diffeomorphic approximation of $W^{1, 1}$ planar Sobolev homeomorphisms, J. Eur. Math. Soc., 20 (2018), 597-656. doi: 10.4171/jems/774.
[51]	T. Iwaniec and F. W. Gehring, The limit of mappings with finite distortion, Ann. Acad. Sci. Fennicae Math., 24 (1999), 253-264.
[52]	T. Iwaniec, Nonlinear Cauchy-Riemann operators in $ \mathbb{R}^n$, Trans. Amer. Math. Soc., 354 (2002), 1961-1995. doi: 10.1090/S0002-9947-02-02914-8.
[53]	T. Iwaniec and J. Onninen, Hyperelastic deformations of smallest total energy, Arch. Ration. Mech. Anal., 194 (2009), 927-986. doi: 10.1007/s00205-008-0192-7.
[54]	T. Iwaniec, L. V. Kovalev and J. Onninen, Approximation up to the boundary of homeomorphisms of finite Dirichlet energy, Bull. London Math. Soc., 44 (2012), 871-881. doi: 10.1112/blms/bds016.
[55]	T. Iwaniec and J. Onninen, Monotone Sobolev mappings of planar domains and surfaces, Arch. Ration. Mech. Anal., 219 (2016), 159-181. doi: 10.1007/s00205-015-0894-6.
[56]	T. Iwaniec and V. Šverák, On mappings with integrable dilatation, Proc. Amer. Math. Soc., 118 (1993), 181-188. doi: 10.1090/S0002-9939-1993-1160301-5.
[57]	P. Koskela and J. Onninen, Mappings of finite distortion: decay of the Jacobian in the plane, Adv. Calc. Var., 1 (2008), 309-321.
[58]	P. Koskela, J. Onninen and K. Rajala, Mappings of finite distortion: decay of the Jacobian, J. Geom. Anal., 22 (2012), 964-976. doi: 10.1007/s12220-011-9224-x.
[59]	J. Kristensen, A necessary and sufficient condition for lower semicontinuity, Nonlinear Anal., 120 (2015), 43-56. doi: 10.1016/j.na.2015.02.018.
[60]	J. Kristensen and B. Raita, Oscillation and concentration in sequences of PDE constrained measures, Arch. Rational Mech. Anal., 246 (2022), 823-875. doi: 10.1007/s00205-022-01828-7.
[61]	R. Luisto and P. Pankka, Stoilow factorization revisited, Expo. Math., 38 (2020), 303-318.
[62]	P. Marcellini, Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals, Manuscr. Math., 51 (1985), 1-28.
[63]	P. Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. Inst. Henri Poincaré (C) Anal. Non Linéaire, 3 (1986), 391-409. doi: 10.1016/s0294-1449(16)30379-1.
[64]	G. Martin and C. Yao, The $L^p$ Teichmüller Theory: Existence and Regu; arity of Critical Points, Arch. Ration. Mech. Anal., 248 (2024), 35 pp.
[65]	N. G. Meyers, Quasi-convexity and lower semi-continuity of multiple variational integrals of any order, Trans. Amer. Math. Soc., 119 (1965), 125-149. doi: 10.1090/S0002-9947-1965-0188838-3.
[66]	C. B. Morrey Jr., Quasi-convexity and the lower semicontinuity of multiple integrals, Pac. J. Math., 2 (1952), 25-53. doi: 10.2140/pjm.1952.2.25.
[67]	S. Müller, Variational models for microstructure and phase transitions, in Calculus of variations and geometric evolution problems, pp. 85-210. Springer, Berlin, Heidelberg, 1999.
[68]	S. Müller and S. J. Spector, An existence theory for nonlinear elasticity that allows for cavitation, Arch. Ration. Mech. Anal., 131 (1995), 1-66.
[69]	S. Müller, J. Sivaloganathan and S. J. Spector, An isoperimetric estimate and $ \mathrm{W}^{1, p}$-quasiconvexity in nonlinear elasticity, Calc. Var., 8 (1999), 159-176.
[70]	F. Nazarov and A. Volberg, Heat extension of the Beurling operator and estimates for its norm, St. Petersburg Math. J., 15 (2004), 563-573. doi: 10.1090/S1061-0022-04-00822-2.
[71]	P. Pedregal and V. Šverák, A note on quasiconvexity and rank-one convexity for $2 \times 2$ matrices, J. Convex Anal., 5 (1998), 107-117.
[72]	S. Petermichl and A. Volberg, Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular, Duke Math. J., 112 (2002), 281-305.
[73]	R. W. Ogden, Non-Linear Elastic Deformations, Dover, 1997.
[74]	R. Ortigosa, D. Ruiz and A. J. Gilc, et al., A stabilisation approach for topology optimisation of hyperelastic structures with the SIMP method, Comput. Methods Appl. Mech. Engrg., 364 (2020), 24 pp.
[75]	P. Pedregal, Jensen's inequality in the calculus of variations, Differ. Integral Equ., 7 (1994), 57-72.
[76]	P. Pedregal, Parametrized Measures and Variational Principles, Birkhäuser, 1997.
[77]	F. Rindler, Calculus of Variations, Springer, 2018.
[78]	M. Sakai, Null quadrature domains, J. d'Analyse Math., 40 (1981), 144-154. doi: 10.1007/BF02790159.
[79]	J. Schröder and P. Neff, Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions, Int. J. Solids Struct., 40 (2003), 401-445. doi: 10.1016/S0020-7683(02)00458-4.
[80]	S. Stoilow, Sur les transformations continues et la topologie des fonctions analytiques, Ann. Sci. Éc. Norm. Supér., 45 (1928), 347-382. doi: 10.24033/asens.787.
[81]	V. Šverák, Examples of rank-one convex functions, Proc. R. Soc. Edinb. A: Math, 114 (1990), 237-242.
[82]	V. Šverák, Rank-one convexity does not imply quasiconvexity, Proc. R. Soc. Edinb. A: Math, 120 (1992), 185-189.
[83]	V. Šverák, Lower-semicontinuity of variational integrals and compensated compactness, Proceedings of the International Congress of Math., vol. 1, no. 2, Zürich, 1994, Birkhäuser, Basel, 1995, pp. 1153-1158.
[84]	J. Voss, R. J. Martin and I.-D. Ghiba, et al., Sharp Rank-One Convexity Conditions in Planar Isotropic Elasticity for the Additive Volumetric-Isochoric Split, J. Elast., 143 (2021), 301-335. doi: 10.1007/s10659-021-09817-9.
[85]	J. Voss, R. J. Martin and I.-D. Ghiba, et al., Morrey's conjecture for the planar volumetricisochoric split: least rank-one convex energy functions, J. Nonlinear Sci., 32 (2022), 1-49.
[86]	J. Voss, R. J. Martin and O. Sander, et al., Numerical approaches for investigating quasiconvexity in the context of Morrey's conjecture, J. Nonlinear Sci., 32 (2022), 41 pp.