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

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

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.
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

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