Estimates for the constant in two nonlinear Korn inequalities. (English) Zbl 07357412
Summary: A nonlinear Korn inequality estimates the distance between two immersions from an open subset of \(\mathbb{R}^n\) into the Euclidean space \(\mathbb{R}^k\), \(k\geqslant n\geqslant 1\), in terms of the distance between specific tensor fields that determine the two immersions up to a rigid motion in \(\mathbb{R}^k\). We establish new inequalities of this type in two cases: when \(k=n\), in which case the tensor fields are the square roots of the metric tensor fields induced by the two immersions, and when \(k=3\) and \(n=2\), in which case the tensor fields are defined in terms of the fundamental forms induced by the immersions. These inequalities have the property that their constants depend only on the open subset over which the immersions are defined and on three scalar parameters defining the regularity of the immersions, instead of constants depending on one of the immersions, considered as fixed, as up to now.
MSC:
| 74-XX | Mechanics of deformable solids |
Keywords:
nonlinear Korn inequalities; nonlinear elasticity; shell theory; differential geometry; surface theoryReferences:
| [1] | Ciarlet, PG, Mardare, C, Mardare, S. Recovery of immersions from their metric tensors and nonlinear Korn inequalities: A brief survey. Chin Ann Math Ser B 2017; 38: 253-280. · Zbl 1362.53051 · doi:10.1007/s11401-016-1070-5 |
| [2] | Ciarlet, PG, Malin, M, Mardare, C. New nonlinear estimates for surfaces in terms of their fundamental forms. C R Acad Sci Paris Ser I 2017; 355: 226-231. · Zbl 1361.53004 · doi:10.1016/j.crma.2017.01.002 |
| [3] | Ciarlet, PG, Mardare, C. Nonlinear Korn inequalities. J Math Pures Appl 2015; 104: 1119-1134. · Zbl 1337.35144 · doi:10.1016/j.matpur.2015.07.007 |
| [4] | Blanchard, D, Griso, G. Decomposition of the deformations of a thin shell. Asymptotic behavior of the Green-St Venant’s strain tensor. J Elasticity 2010; 101: 179-205. · Zbl 1258.74137 · doi:10.1007/s10659-010-9255-8 |
| [5] | Ciarlet, PG, Gratie, L, Mardare, C. A nonlinear Korn inequality on a surface. J Math Pures Appl 2006; 85: 2-16. · Zbl 1094.53001 · doi:10.1016/j.matpur.2005.10.010 |
| [6] | Malin, M, Mardare, C. Nonlinear Korn inequalities on a hypersurface. Chin Ann Math Ser B 2018; 39: 513-534. · Zbl 1396.53009 · doi:10.1007/s11401-018-0080-x |
| [7] | Malin, M, Mardare, C. Nonlinear estimates for hypersurfaces in terms of their fundamental forms. C R Acad Sci Paris Ser I 2017; 355: 1196-1200. · Zbl 1388.53009 · doi:10.1016/j.crma.2017.10.014 |
| [8] | Adams, RA. Sobolev Spaces. New York: Academic Press, 1975. |
| [9] | Maz’ya, V. Sobolev Spaces. Heidelberg: Springer, 1985. |
| [10] | Nečas, J. Les Méthodes Directes en Théorie des Equations Elliptiques. Paris: Masson, 1967. · Zbl 1225.35003 |
| [11] | Anicic, S, Le Dret, H, Raoult, A. The infinitesimal rigid displacement lemma in Lipschitz coordinates and application to shells with minimal regularity. Math Meth Appl Sci 2004; 27: 1283-1299. · Zbl 1156.35477 · doi:10.1002/mma.501 |
| [12] | Ciarlet, PG, Mardare, C. Recovery of a manifold with boundary and its continuity as a function of its metric tensor. J Math Pures Appl 2004; 83: 811-843. · Zbl 1088.74014 · doi:10.1016/j.matpur.2004.01.004 |
| [13] | Whitney, H. Analytic extensions of differentiable functions defined in closed sets. Trans Amer Math Soc 1934; 36: 63-89. · JFM 60.0217.01 · doi:10.1090/S0002-9947-1934-1501735-3 |
| [14] | Ciarlet, PG, Mardare, C. Continuity of a deformation in \(H^1\) as a function of its Cauchy-Green tensor in \(L^1\). J Nonlinear Sci 2004; 14: 415-427. · Zbl 1084.53063 · doi:10.1007/s00332-004-0624-y |
| [15] | Ciarlet, PG. Linear and Nonlinear Functional Analysis with Applications. Philadelphia, PA: SIAM, 2013. · Zbl 1293.46001 |
| [16] | Friesecke, G, James, RD, Müller, S. A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity. Comm Pure Appl Math 2002; 55: 1461-1506. · Zbl 1021.74024 · doi:10.1002/cpa.10048 |
| [17] | Conti, S. Low-energy Deformations of Thin Elastic Plates: Isometric Embeddings and Branching Patterns. Habilitationsschrift, Universität Leipzig, Germany, 2004. · Zbl 1137.74004 |
| [18] | Friesecke, G, James, RD, Müller, S. A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence. Arch Rat Mech Anal 2006; 180: 183-236. · Zbl 1100.74039 · doi:10.1007/s00205-005-0400-7 |
| [19] | Gurtin, ME. An Introduction to Continuum Mechanics. New York: Academic Press, 1981. · Zbl 0559.73001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
