American Institute of Mathematical Sciences

Advanced
June  2012, 5(3): 473-483. doi: 10.3934/dcdss.2012.5.473

Korn's inequalities: The linear vs. the nonlinear case

Philippe Ciarlet 1,

1. 

City University of Hong Kong, Department of Mathematics, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong, China

Received  September 2010 Published  October 2011

It is well known that the linear Korn inequality pervades the theory of three-dimensional linearized elasticity. It is thus conceivable that nonlinear Korn's inequalities could likewise play a role in the theory of three-dimensional nonlinear elasticity. In this paper, we describe the (available to this date) linear and nonlinear Korn's inequalities and we discuss the resemblances, but also the sometimes intriguing differences, that exist between these two kinds of inequalities.
Keywords: multifractal analysis., Poincaré recurrences, Dimension theory.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Philippe Ciarlet. Korn's inequalities: The linear vs. the nonlinear case. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 473-483. doi: 10.3934/dcdss.2012.5.473
References:
[1]

S. S. Antman, Ordinary differential equations of non-linear elasticity. I. Foundations of the theories of nonlinearly elastic rods and shells,, Arch. Rational Mech. Anal., 61 (1976), 307.  doi: 10.1007/BF00250722.  Google Scholar

[2]

J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity,, Arch. Rational Mech. Anal., 63 (): 337.  doi: 10.1007/BF00279992.  Google Scholar

[3]

P. G. Ciarlet, "Mathematical Elasticity. Volume I. Three-Dimensional Elasticity,'', Studies in Mathematics and its Applications, 20 (1988).   Google Scholar

[4]

P. G. Ciarlet, "An Introduction to Differential Geometry with Applications to Elasticity,'', Reprinted from J. Elasticity, 78/79 (2005).   Google Scholar

[5]

P. G. Ciarlet and P. Ciarlet, Jr., Another approach to linearized elasticity and a new proof of Korn's inequality,, Math. Models Methods Appl. Sci., 15 (2005), 259.  doi: 10.1142/S0218202505000352.  Google Scholar

[6]

P. G. Ciarlet and F. Laurent, Continuity of a deformation as a function of its Cauchy-Green tensor,, Arch. Ration. Mech. Anal., 167 (2003), 255.  doi: 10.1007/s00205-003-0246-9.  Google Scholar

[7]

P. G. Ciarlet and C. Mardare, On rigid and infinitesimal rigid displacements in three-dimensional elasticity,, Math. Models Methods Appl. Sci., 13 (2003), 1589.  doi: 10.1142/S0218202503003045.  Google Scholar

[8]

P. G. Ciarlet and C. Mardare, Recovery of a manifold with boundary and its continuity as a function of its metric tensor,, J. Math. Pures Appl. (9), 83 (2004), 811.  doi: 10.1016/j.matpur.2004.01.004.  Google Scholar

[9]

P. G. Ciarlet and C. Mardare, Continuity of a deformation in $H^1$ as a function of its Cauchy-Green tensor in $L^1$,, J. Nonlinear Sci., 14 (2004), 415.  doi: 10.1007/s00332-004-0624-y.  Google Scholar

[10]

P. G. Ciarlet and C. Mardare, Existence theorems in intrinsic nonlinear elasticity,, J. Math. Pures Appl., 94 (2010), 229.  doi: 10.1016/j.matpur.2010.02.002.  Google Scholar

[11]

P. G. Ciarlet and C. Mardare, Remarks on Korn's inequalities in $W^{1,p} (\Omega)$,, in preparation., ().   Google Scholar

[12]

P. G. Ciarlet, C. Mardare and M. Shen, Saint Venant compatibility equations in curvilinear coordinates,, Analysis and Applications (Singap.), 5 (2007), 231.  doi: 10.1142/S0219530507000973.  Google Scholar

[13]

S. Conti, "Low-energy Deformations of Thin Elastic Plates: Isometric Embeddings and Branching Patterns,'', Habilitationsschrift, (2004).   Google Scholar

[14]

G. Duvaut and J.-L. Lions, "Les Inéquations en Mécanique et en Physique,'', Travaux et Recherches Mathématiques, (1972).   Google Scholar

[15]

K. O. Friedrichs, On the boundary-value problems of the theory of elasticity and Korn's inequality,, Ann. of Math. (2), 48 (1947), 441.  doi: 10.2307/1969180.  Google Scholar

[16]

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.  doi: 10.1002/cpa.10048.  Google Scholar

[17]

G. Geymonat and P. Suquet, Functional spaces for Norton-Hoff materials,, Math. Models Methods Appl. Sci., 8 (1986), 206.   Google Scholar

[18]

J. Gobert, Une inégalité fondamentale de la théorie de l'élasticité,, Bull. Soc. Roy. Sci. Liège, 31 (1962), 182.   Google Scholar

[19]

C. O. Horgan, Korn's inequalities and their applications in continuum mechanics,, SIAM Review, 37 (1995), 491.  doi: 10.1137/1037123.  Google Scholar

[20]

F. John, Rotation and strain,, Comm. Pure Appl. Math., 14 (1961), 391.  doi: 10.1002/cpa.3160140316.  Google Scholar

[21]

F. John, Bounds for deformations in terms of average strains,, in, (1972), 129.   Google Scholar

[22]

R. V. Kohn, New integral estimates for deformations in terms of their nonlinear strains,, Arch. Rational Mech. Anal., 78 (1982), 131.  doi: 10.1007/BF00250837.  Google Scholar

[23]

A. Korn, Die Eigenschwingungen eines elastischen Körpers mit ruhender Oberfläche,, Sitzungsberichte der Mathematisch-physikalischen Klasse der Königlich bayerischen Akademie der Wissenschaften zu München, 36 (1906), 351.   Google Scholar

[24]

C. Mardare, On the recovery of a manifold with prescribed metric tensor,, Analysis and Applications (Singap.), 1 (2003), 433.  doi: 10.1142/S0219530503000235.  Google Scholar

[25]

S. Mardare, Inequality of Korn's type on compact surfaces without boundary,, Chinese Annals Math. Ser. B, 24 (2003), 191.  doi: 10.1142/S0252959903000177.  Google Scholar

[26]

S. Mardare, On isometric immersions of a Riemannian space with little regularity,, Analysis and Applications (Singap.), 2 (2004), 193.  doi: 10.1142/S0219530504000357.  Google Scholar

[27]

S. Mardare, On Pfaff systems with $L^p$ coefficients and their applications in differential geometry,, J. Math. Pures Appl. (9), 84 (2005), 1659.  doi: 10.1016/j.matpur.2005.08.002.  Google Scholar

[28]

S. Mardare, On systems of first order linear partial differential equations with $L^p$ coefficients,, Advances in Differential Equations, 12 (2007), 301.   Google Scholar

[29]

Y. Reshetnyak, Mappings of domains in $\mathbbR^n$ and their metric tensors,, Siberian Math. J., 44 (2003), 332.  doi: 10.1023/A:1022945123237.  Google Scholar

show all references

References:
[1]

S. S. Antman, Ordinary differential equations of non-linear elasticity. I. Foundations of the theories of nonlinearly elastic rods and shells,, Arch. Rational Mech. Anal., 61 (1976), 307.  doi: 10.1007/BF00250722.  Google Scholar

[2]

J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity,, Arch. Rational Mech. Anal., 63 (): 337.  doi: 10.1007/BF00279992.  Google Scholar

[3]

P. G. Ciarlet, "Mathematical Elasticity. Volume I. Three-Dimensional Elasticity,'', Studies in Mathematics and its Applications, 20 (1988).   Google Scholar

[4]

P. G. Ciarlet, "An Introduction to Differential Geometry with Applications to Elasticity,'', Reprinted from J. Elasticity, 78/79 (2005).   Google Scholar

[5]

P. G. Ciarlet and P. Ciarlet, Jr., Another approach to linearized elasticity and a new proof of Korn's inequality,, Math. Models Methods Appl. Sci., 15 (2005), 259.  doi: 10.1142/S0218202505000352.  Google Scholar

[6]

P. G. Ciarlet and F. Laurent, Continuity of a deformation as a function of its Cauchy-Green tensor,, Arch. Ration. Mech. Anal., 167 (2003), 255.  doi: 10.1007/s00205-003-0246-9.  Google Scholar

[7]

P. G. Ciarlet and C. Mardare, On rigid and infinitesimal rigid displacements in three-dimensional elasticity,, Math. Models Methods Appl. Sci., 13 (2003), 1589.  doi: 10.1142/S0218202503003045.  Google Scholar

[8]

P. G. Ciarlet and C. Mardare, Recovery of a manifold with boundary and its continuity as a function of its metric tensor,, J. Math. Pures Appl. (9), 83 (2004), 811.  doi: 10.1016/j.matpur.2004.01.004.  Google Scholar

[9]

P. G. Ciarlet and C. Mardare, Continuity of a deformation in $H^1$ as a function of its Cauchy-Green tensor in $L^1$,, J. Nonlinear Sci., 14 (2004), 415.  doi: 10.1007/s00332-004-0624-y.  Google Scholar

[10]

P. G. Ciarlet and C. Mardare, Existence theorems in intrinsic nonlinear elasticity,, J. Math. Pures Appl., 94 (2010), 229.  doi: 10.1016/j.matpur.2010.02.002.  Google Scholar

[11]

P. G. Ciarlet and C. Mardare, Remarks on Korn's inequalities in $W^{1,p} (\Omega)$,, in preparation., ().   Google Scholar

[12]

P. G. Ciarlet, C. Mardare and M. Shen, Saint Venant compatibility equations in curvilinear coordinates,, Analysis and Applications (Singap.), 5 (2007), 231.  doi: 10.1142/S0219530507000973.  Google Scholar

[13]

S. Conti, "Low-energy Deformations of Thin Elastic Plates: Isometric Embeddings and Branching Patterns,'', Habilitationsschrift, (2004).   Google Scholar

[14]

G. Duvaut and J.-L. Lions, "Les Inéquations en Mécanique et en Physique,'', Travaux et Recherches Mathématiques, (1972).   Google Scholar

[15]

K. O. Friedrichs, On the boundary-value problems of the theory of elasticity and Korn's inequality,, Ann. of Math. (2), 48 (1947), 441.  doi: 10.2307/1969180.  Google Scholar

[16]

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.  doi: 10.1002/cpa.10048.  Google Scholar

[17]

G. Geymonat and P. Suquet, Functional spaces for Norton-Hoff materials,, Math. Models Methods Appl. Sci., 8 (1986), 206.   Google Scholar

[18]

J. Gobert, Une inégalité fondamentale de la théorie de l'élasticité,, Bull. Soc. Roy. Sci. Liège, 31 (1962), 182.   Google Scholar

[19]

C. O. Horgan, Korn's inequalities and their applications in continuum mechanics,, SIAM Review, 37 (1995), 491.  doi: 10.1137/1037123.  Google Scholar

[20]

F. John, Rotation and strain,, Comm. Pure Appl. Math., 14 (1961), 391.  doi: 10.1002/cpa.3160140316.  Google Scholar

[21]

F. John, Bounds for deformations in terms of average strains,, in, (1972), 129.   Google Scholar

[22]

R. V. Kohn, New integral estimates for deformations in terms of their nonlinear strains,, Arch. Rational Mech. Anal., 78 (1982), 131.  doi: 10.1007/BF00250837.  Google Scholar

[23]

A. Korn, Die Eigenschwingungen eines elastischen Körpers mit ruhender Oberfläche,, Sitzungsberichte der Mathematisch-physikalischen Klasse der Königlich bayerischen Akademie der Wissenschaften zu München, 36 (1906), 351.   Google Scholar

[24]

C. Mardare, On the recovery of a manifold with prescribed metric tensor,, Analysis and Applications (Singap.), 1 (2003), 433.  doi: 10.1142/S0219530503000235.  Google Scholar

[25]

S. Mardare, Inequality of Korn's type on compact surfaces without boundary,, Chinese Annals Math. Ser. B, 24 (2003), 191.  doi: 10.1142/S0252959903000177.  Google Scholar

[26]

S. Mardare, On isometric immersions of a Riemannian space with little regularity,, Analysis and Applications (Singap.), 2 (2004), 193.  doi: 10.1142/S0219530504000357.  Google Scholar

[27]

S. Mardare, On Pfaff systems with $L^p$ coefficients and their applications in differential geometry,, J. Math. Pures Appl. (9), 84 (2005), 1659.  doi: 10.1016/j.matpur.2005.08.002.  Google Scholar

[28]

S. Mardare, On systems of first order linear partial differential equations with $L^p$ coefficients,, Advances in Differential Equations, 12 (2007), 301.   Google Scholar

[29]

Y. Reshetnyak, Mappings of domains in $\mathbbR^n$ and their metric tensors,, Siberian Math. J., 44 (2003), 332.  doi: 10.1023/A:1022945123237.  Google Scholar

[1]

V. Afraimovich, J. Schmeling, Edgardo Ugalde, Jesús Urías. Spectra of dimensions for Poincaré recurrences. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 901-914. doi: 10.3934/dcds.2000.6.901
[2]

B. Fernandez, E. Ugalde, J. Urías. Spectrum of dimensions for Poincaré recurrences of Markov maps. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 835-849. doi: 10.3934/dcds.2002.8.835
[3]

Juan Wang, Xiaodan Zhang, Yun Zhao. Dimension estimates for arbitrary subsets of limit sets of a Markov construction and related multifractal analysis. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2315-2332. doi: 10.3934/dcds.2014.34.2315
[4]

Godofredo Iommi, Bartłomiej Skorulski. Multifractal analysis for the exponential family. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 857-869. doi: 10.3934/dcds.2006.16.857
[5]

V. Afraimovich, Jean-René Chazottes, Benoît Saussol. Pointwise dimensions for Poincaré recurrences associated with maps and special flows. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 263-280. doi: 10.3934/dcds.2003.9.263
[6]

Julien Barral, Yan-Hui Qu. On the higher-dimensional multifractal analysis. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 1977-1995. doi: 10.3934/dcds.2012.32.1977
[7]

Mario Roy, Mariusz Urbański. Multifractal analysis for conformal graph directed Markov systems. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 627-650. doi: 10.3934/dcds.2009.25.627
[8]

Zhihui Yuan. Multifractal analysis of random weak Gibbs measures. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5367-5405. doi: 10.3934/dcds.2017234
[9]

Luis Barreira. Dimension theory of flows: A survey. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3345-3362. doi: 10.3934/dcdsb.2015.20.3345
[10]

Luis Barreira, César Silva. Lyapunov exponents for continuous transformations and dimension theory. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 469-490. doi: 10.3934/dcds.2005.13.469
[11]

Valentin Afraimovich, Jean-Rene Chazottes and Benoit Saussol. Local dimensions for Poincare recurrences. Electronic Research Announcements, 2000, 6: 64-74.

[12]

Zied Douzi, Bilel Selmi. On the mutual singularity of multifractal measures. Electronic Research Archive, 2020, 28 (1) : 423-432. doi: 10.3934/era.2020024
[13]

Mirela Domijan, Markus Kirkilionis. Graph theory and qualitative analysis of reaction networks. Networks & Heterogeneous Media, 2008, 3 (2) : 295-322. doi: 10.3934/nhm.2008.3.295
[14]

Jean-Pierre Francoise, Claude Piquet. Global recurrences of multi-time scaled systems. Conference Publications, 2011, 2011 (Special) : 430-436. doi: 10.3934/proc.2011.2011.430
[15]

Jerrold E. Marsden, Alexey Tret'yakov. Factor analysis of nonlinear mappings: p-regularity theory. Communications on Pure & Applied Analysis, 2003, 2 (4) : 425-445. doi: 10.3934/cpaa.2003.2.425
[16]

Lars Olsen. First return times: multifractal spectra and divergence points. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 635-656. doi: 10.3934/dcds.2004.10.635
[17]

Imen Bhouri, Houssem Tlili. On the multifractal formalism for Bernoulli products of invertible matrices. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1129-1145. doi: 10.3934/dcds.2009.24.1129
[18]

Eva Miranda, Romero Solha. A Poincaré lemma in geometric quantisation. Journal of Geometric Mechanics, 2013, 5 (4) : 473-491. doi: 10.3934/jgm.2013.5.473
[19]

Jiahang Che, Li Chen, Simone GÖttlich, Anamika Pandey, Jing Wang. Boundary layer analysis from the Keller-Segel system to the aggregation system in one space dimension. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1013-1036. doi: 10.3934/cpaa.2017049
[20]

Vaughn Climenhaga. Multifractal formalism derived from thermodynamics for general dynamical systems. Electronic Research Announcements, 2010, 17: 1-11. doi: 10.3934/era.2010.17.1

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences