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

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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

Abstract
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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 and 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), 307-351. doi: 10.1007/BF00250722.
[2]	J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity,, Arch. Rational Mech. Anal., 63 (): 337. doi: 10.1007/BF00279992.
[3]	P. G. Ciarlet, "Mathematical Elasticity. Volume I. Three-Dimensional Elasticity,'' Studies in Mathematics and its Applications, 20, North-Holland Publishing Co., Amsterdam, 1988.
[4]	P. G. Ciarlet, "An Introduction to Differential Geometry with Applications to Elasticity,'' Reprinted from J. Elasticity, 78/79 (2005), Springer, Dordrecht, 2005.
[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), 259-271. doi: 10.1142/S0218202505000352.
[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), 255-269. doi: 10.1007/s00205-003-0246-9.
[7]	P. G. Ciarlet and C. Mardare, On rigid and infinitesimal rigid displacements in three-dimensional elasticity, Math. Models Methods Appl. Sci., 13 (2003), 1589-1598. doi: 10.1142/S0218202503003045.
[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), 811-843. doi: 10.1016/j.matpur.2004.01.004.
[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), 415-427. doi: 10.1007/s00332-004-0624-y.
[10]	P. G. Ciarlet and C. Mardare, Existence theorems in intrinsic nonlinear elasticity, J. Math. Pures Appl., 94 (2010), 229-243. doi: 10.1016/j.matpur.2010.02.002.
[11]	P. G. Ciarlet and C. Mardare, Remarks on Korn's inequalities in $W^{1,p} (\Omega)$,, in preparation., ().
[12]	P. G. Ciarlet, C. Mardare and M. Shen, Saint Venant compatibility equations in curvilinear coordinates, Analysis and Applications (Singap.), 5 (2007), 231-251. doi: 10.1142/S0219530507000973.
[13]	S. Conti, "Low-energy Deformations of Thin Elastic Plates: Isometric Embeddings and Branching Patterns,'' Habilitationsschrift, Universität Leipzig, 2004.
[14]	G. Duvaut and J.-L. Lions, "Les Inéquations en Mécanique et en Physique,'' Travaux et Recherches Mathématiques, No. 21, Dunod, Paris, 1972.
[15]	K. O. Friedrichs, On the boundary-value problems of the theory of elasticity and Korn's inequality, Ann. of Math. (2), 48 (1947), 441-471. doi: 10.2307/1969180.
[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), 1461-1506. doi: 10.1002/cpa.10048.
[17]	G. Geymonat and P. Suquet, Functional spaces for Norton-Hoff materials, Math. Models Methods Appl. Sci., 8 (1986), 206-222.
[18]	J. Gobert, Une inégalité fondamentale de la théorie de l'élasticité, Bull. Soc. Roy. Sci. Liège, 31 (1962), 182-191.
[19]	C. O. Horgan, Korn's inequalities and their applications in continuum mechanics, SIAM Review, 37 (1995), 491-511. doi: 10.1137/1037123.
[20]	F. John, Rotation and strain, Comm. Pure Appl. Math., 14 (1961), 391-413. doi: 10.1002/cpa.3160140316.
[21]	F. John, Bounds for deformations in terms of average strains, in "Inequalities III'' (ed. O. Shisha), Academic Press, New York, (1972), 129-144.
[22]	R. V. Kohn, New integral estimates for deformations in terms of their nonlinear strains, Arch. Rational Mech. Anal., 78 (1982), 131-172. doi: 10.1007/BF00250837.
[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), 351-402.
[24]	C. Mardare, On the recovery of a manifold with prescribed metric tensor, Analysis and Applications (Singap.), 1 (2003), 433-453. doi: 10.1142/S0219530503000235.
[25]	S. Mardare, Inequality of Korn's type on compact surfaces without boundary, Chinese Annals Math. Ser. B, 24 (2003), 191-204. doi: 10.1142/S0252959903000177.
[26]	S. Mardare, On isometric immersions of a Riemannian space with little regularity, Analysis and Applications (Singap.), 2 (2004), 193-226. doi: 10.1142/S0219530504000357.
[27]	S. Mardare, On Pfaff systems with $L^p$ coefficients and their applications in differential geometry, J. Math. Pures Appl. (9), 84 (2005), 1659-1692. doi: 10.1016/j.matpur.2005.08.002.
[28]	S. Mardare, On systems of first order linear partial differential equations with $L^p$ coefficients, Advances in Differential Equations, 12 (2007), 301-360.
[29]	Y. Reshetnyak, Mappings of domains in $\mathbbR^n$ and their metric tensors, Siberian Math. J., 44 (2003), 332-345. doi: 10.1023/A:1022945123237.

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), 307-351. doi: 10.1007/BF00250722.
[2]	J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity,, Arch. Rational Mech. Anal., 63 (): 337. doi: 10.1007/BF00279992.
[3]	P. G. Ciarlet, "Mathematical Elasticity. Volume I. Three-Dimensional Elasticity,'' Studies in Mathematics and its Applications, 20, North-Holland Publishing Co., Amsterdam, 1988.
[4]	P. G. Ciarlet, "An Introduction to Differential Geometry with Applications to Elasticity,'' Reprinted from J. Elasticity, 78/79 (2005), Springer, Dordrecht, 2005.
[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), 259-271. doi: 10.1142/S0218202505000352.
[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), 255-269. doi: 10.1007/s00205-003-0246-9.
[7]	P. G. Ciarlet and C. Mardare, On rigid and infinitesimal rigid displacements in three-dimensional elasticity, Math. Models Methods Appl. Sci., 13 (2003), 1589-1598. doi: 10.1142/S0218202503003045.
[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), 811-843. doi: 10.1016/j.matpur.2004.01.004.
[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), 415-427. doi: 10.1007/s00332-004-0624-y.
[10]	P. G. Ciarlet and C. Mardare, Existence theorems in intrinsic nonlinear elasticity, J. Math. Pures Appl., 94 (2010), 229-243. doi: 10.1016/j.matpur.2010.02.002.
[11]	P. G. Ciarlet and C. Mardare, Remarks on Korn's inequalities in $W^{1,p} (\Omega)$,, in preparation., ().
[12]	P. G. Ciarlet, C. Mardare and M. Shen, Saint Venant compatibility equations in curvilinear coordinates, Analysis and Applications (Singap.), 5 (2007), 231-251. doi: 10.1142/S0219530507000973.
[13]	S. Conti, "Low-energy Deformations of Thin Elastic Plates: Isometric Embeddings and Branching Patterns,'' Habilitationsschrift, Universität Leipzig, 2004.
[14]	G. Duvaut and J.-L. Lions, "Les Inéquations en Mécanique et en Physique,'' Travaux et Recherches Mathématiques, No. 21, Dunod, Paris, 1972.
[15]	K. O. Friedrichs, On the boundary-value problems of the theory of elasticity and Korn's inequality, Ann. of Math. (2), 48 (1947), 441-471. doi: 10.2307/1969180.
[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), 1461-1506. doi: 10.1002/cpa.10048.
[17]	G. Geymonat and P. Suquet, Functional spaces for Norton-Hoff materials, Math. Models Methods Appl. Sci., 8 (1986), 206-222.
[18]	J. Gobert, Une inégalité fondamentale de la théorie de l'élasticité, Bull. Soc. Roy. Sci. Liège, 31 (1962), 182-191.
[19]	C. O. Horgan, Korn's inequalities and their applications in continuum mechanics, SIAM Review, 37 (1995), 491-511. doi: 10.1137/1037123.
[20]	F. John, Rotation and strain, Comm. Pure Appl. Math., 14 (1961), 391-413. doi: 10.1002/cpa.3160140316.
[21]	F. John, Bounds for deformations in terms of average strains, in "Inequalities III'' (ed. O. Shisha), Academic Press, New York, (1972), 129-144.
[22]	R. V. Kohn, New integral estimates for deformations in terms of their nonlinear strains, Arch. Rational Mech. Anal., 78 (1982), 131-172. doi: 10.1007/BF00250837.
[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), 351-402.
[24]	C. Mardare, On the recovery of a manifold with prescribed metric tensor, Analysis and Applications (Singap.), 1 (2003), 433-453. doi: 10.1142/S0219530503000235.
[25]	S. Mardare, Inequality of Korn's type on compact surfaces without boundary, Chinese Annals Math. Ser. B, 24 (2003), 191-204. doi: 10.1142/S0252959903000177.
[26]	S. Mardare, On isometric immersions of a Riemannian space with little regularity, Analysis and Applications (Singap.), 2 (2004), 193-226. doi: 10.1142/S0219530504000357.
[27]	S. Mardare, On Pfaff systems with $L^p$ coefficients and their applications in differential geometry, J. Math. Pures Appl. (9), 84 (2005), 1659-1692. doi: 10.1016/j.matpur.2005.08.002.
[28]	S. Mardare, On systems of first order linear partial differential equations with $L^p$ coefficients, Advances in Differential Equations, 12 (2007), 301-360.
[29]	Y. Reshetnyak, Mappings of domains in $\mathbbR^n$ and their metric tensors, Siberian Math. J., 44 (2003), 332-345. doi: 10.1023/A:1022945123237.

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