Geometric rigidity of conformal matrices

Faraco, Daniel; Zhong, Xiao

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 4, pp. 557-585.

We provide a geometric rigidity estimate à la Friesecke-James-Müller for conformal matrices. Namely, we replace $\mathrm{SO}\left(n\right)$ by an arbitrary compact set of conformal matrices, bounded away from $0$ and invariant under $\mathrm{SO}\left(n\right)$, and rigid motions by Möbius transformations.

[1] A. F. Beardon, “The Geometry of Discrete Groups”, Graduate Texts in Mathematics, Vol. 91, Springer-Verlag, New York, 1995. Corrected reprint of the 1983 original. | MR | Zbl

[2] N. Chaudhuri and S. Müller, Rigidity estimate for two incompatible wells, Calc. Var. Partial Differential Equations 19 (2004), 379-390. | MR | Zbl

[3] G. Dal Maso, M. Negri and D. Percivale, Linearized elasticity as $\Gamma$-limit of finite elasticity. Calculus of variations, nonsmooth analysis and related topics, Set-Valued Anal. 10 (2002), 165-183. | MR | Zbl

[4] E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25-43. | MR | Zbl

[5] D. Faraco, Rigidity of sets of matrices, In: “Future Trends in Geometric Function Theory”, Rep. Univ. Jyväskylä Dep. Math. Stat., Vol. 92, Univ. Jyväskylä, Jyväskylä, 2003, 77-83. | MR | Zbl

[6] G. Friesecke, R. D. James, M. G. Mora and S. Müller, Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence, C. R. Math. Acad. Sci. Paris 336 (2003), 697-702. | MR | Zbl

[7] G. Friesecke, S. Müller and R. D. James, Rigorous derivation of nonlinear plate theory and geometric rigidity, C. R. Math. Acad. Sci. Paris 334 (2002), 173-178. | MR | Zbl

[8] G. Friesecke, R. D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math. 55 (2002), 1461-1506. | MR | Zbl

[9] F. W. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), 353-393. | MR | Zbl

[10] E. Giusti, “Direct Methods in the Calculus of Variations”, World Scientific Publishing Co. Inc., River Edge, NJ, 2003. | MR | Zbl

[11] J. Heinonen, T. Kilpeläinen and O. Martio, “Nonlinear Potential Theory of Degenerate Elliptic Equations”, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1993. Oxford Science Publications. | MR | Zbl

[12] S. Hencl, P. Koskela and X. Zhong, Mappings of finite distortion: Reverse inequalities for the $\mathrm{Jacobian}$, Preprint 310 at the University of Jyväskylä (2004). | MR

[13] T. Iwaniec, $p$-harmonic tensors and quasiregular mappings, Ann. of Math. (2) 136 (1992), 589-624. | MR | Zbl

[14] T. Iwaniec and G. Martin, “Geometric Function Theory and Non-Linear Analysis”, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 2001. | MR | Zbl

[15] F. John, Rotation and strain, Comm. Pure Appl. Math. 14 (1961), 391-413. | MR | Zbl

[16] J. Liouville, Théorème sur l’équation $d{x}^2+d{y}^2+d{z}^2=\lambda (d{\alpha }^2+d{\beta }^2+d{\gamma }^2$, J. Math. Pures Appl. 15 (1850).

[17] M. G. Mora and S. Müller, Derivation of the nonlinear bending-torsion theory for inextensible rods by $\Gamma$-convergence, Calc. Var. Partial Differential Equations 18 (2003), 287-305. | MR | Zbl

[18] M. G. Mora and S. Müller, A nonlinear model for inextensible rods as a low energy $\Gamma$-limit of three-dimensional nonlinear elasticity, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), 271-293. | Numdam | MR | Zbl

[19] J. G. Rešetnjak, Liouville's conformal mapping theorem under minimal regularity hypotheses, Sibirsk. Mat. Zh. 8 (1967), 835-840. | MR | Zbl

[20] J. G. Rešetnjak, Estimates for certain differential operators with finite-dimensional kernel, Sibirsk. Mat. Zh. 11 (1970), 414-428. | MR | Zbl

[21] J. G. Rešetnjak, “Stability Theorems in Geometry and Analysis”, Mathematics and its Applications, Vol. 304, Kluwer Academic Publishers Group, Dordrecht, 1994. Translated from the 1982 Russian original by N. S. Dairbekov and V. N. Dyatlov, and revised by the author, Translation edited and with a foreword by S. S. Kutateladze. | MR | Zbl