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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 |
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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