Weighted asymptotic Korn and interpolation Korn inequalities with singular weights
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- by Davit Harutyunyan and Hayk Mikayelyan PDF
- Proc. Amer. Math. Soc. 147 (2019), 3635-3647 Request permission
Abstract:
In this work we derive asymptotically sharp weighted Korn and Korn-like interpolation (or first and a half) inequalities in thin domains with singular weights. The constants $K$ (Korn’s constant) in the inequalities depend on the domain thickness $h$ according to a power rule $K=Ch^\alpha ,$ where $C>0$ and $\alpha \in R$ are constants independent of $h$ and the displacement field. The sharpness of the estimates is understood in the sense that the asymptotics $h^\alpha$ is optimal as $h\to 0.$ The choice of the weights is motivated by several factors; in particular a spatial case occurs when making Cartesian to polar change of variables in two dimensions.References
- Gabriel Acosta, María E. Cejas, and Ricardo G. Durán, Improved Poincaré inequalities and solutions of the divergence in weighted forms, Ann. Acad. Sci. Fenn. Math. 42 (2017), no. 1, 211–226. MR 3558524, DOI 10.5186/aasfm.2017.4212
- Gabriel Acosta and Ricardo G. Durán, Divergence operator and related inequalities, SpringerBriefs in Mathematics, Springer, New York, 2017. MR 3618122, DOI 10.1007/978-1-4939-6985-2
- Gabriel Acosta, Ricardo G. Durán, and Fernando López García, Korn inequality and divergence operator: counterexamples and optimality of weighted estimates, Proc. Amer. Math. Soc. 141 (2013), no. 1, 217–232. MR 2988724, DOI 10.1090/S0002-9939-2012-11408-X
- Gabriel Acosta, Ricardo G. Durán, and Ariel L. Lombardi, Weighted Poincaré and Korn inequalities for Hölder $\alpha$ domains, Math. Methods Appl. Sci. 29 (2006), no. 4, 387–400. MR 2198138, DOI 10.1002/mma.680
- 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. MR 22750, DOI 10.2307/1969180
- Gero Friesecke, Richard D. James, and Stefan Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math. 55 (2002), no. 11, 1461–1506. MR 1916989, DOI 10.1002/cpa.10048
- Gero Friesecke, Richard D. James, and Stefan Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence, Arch. Ration. Mech. Anal. 180 (2006), no. 2, 183–236. MR 2210909, DOI 10.1007/s00205-005-0400-7
- Yury Grabovsky and Davit Harutyunyan, Exact scaling exponents in Korn and Korn-type inequalities for cylindrical shells, SIAM J. Math. Anal. 46 (2014), no. 5, 3277–3295. MR 3262603, DOI 10.1137/130948999
- Yury Grabovsky and Davit Harutyunyan, Rigorous derivation of the formula for the buckling load in axially compressed circular cylindrical shells, J. Elasticity 120 (2015), no. 2, 249–276. MR 3367700, DOI 10.1007/s10659-015-9513-x
- Yury Grabovsky and Davit Harutyunyan, Scaling instability in buckling of axially compressed cylindrical shells, J. Nonlinear Sci. 26 (2016), no. 1, 83–119. MR 3441274, DOI 10.1007/s00332-015-9270-9
- Yury Grabovsky and Davit Harutyunyan, Korn inequalities for shells with zero Gaussian curvature, Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 (2018), no. 1, 267–282. MR 3739933, DOI 10.1016/j.anihpc.2017.04.004
- Yury Grabovsky and Lev Truskinovsky, The flip side of buckling, Contin. Mech. Thermodyn. 19 (2007), no. 3-4, 211–243. MR 2326996, DOI 10.1007/s00161-007-0044-y
- Davit Harutyunyan, New asymptotically sharp Korn and Korn-like inequalities in thin domains, J. Elasticity 117 (2014), no. 1, 95–109. MR 3256827, DOI 10.1007/s10659-013-9466-x
- Davit Harutyunyan, Sharp weighted Korn and Korn-like inequalities and an application to washers, J. Elasticity 127 (2017), no. 1, 59–77. MR 3606783, DOI 10.1007/s10659-016-9596-z
- Davit Harutyunyan, Gaussian curvature as an identifier of shell rigidity, Arch. Ration. Mech. Anal. 226 (2017), no. 2, 743–766. MR 3687880, DOI 10.1007/s00205-017-1143-y
- Vladimir Alexandrovich Kondratiev and Olga Arsenievna Oleinik, On Korn’s inequalities, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), no. 16, 483–487 (English, with French summary). MR 995908
- V. A. Kondrat′ev and O. A. Oleĭnik, Boundary value problems for a system in elasticity theory in unbounded domains. Korn inequalities, Uspekhi Mat. Nauk 43 (1988), no. 5(263), 55–98, 239 (Russian); English transl., Russian Math. Surveys 43 (1988), no. 5, 65–119. MR 971465, DOI 10.1070/RM1988v043n05ABEH001945
- A. Korn, Solution générale du problème d’équilibre dans la théorie de l’élasticité, dans le cas ou les efforts sont donnés à la surface, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (2) 10 (1908), 165–269 (French). MR 1508302, DOI 10.5802/afst.251
- A. Korn, Über einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen, Bull. Int. Cracovie Akademie Umiejet, Classe des Sci. Math. Nat. (1909), 705–724.
- Fernando López-García, Weighted Korn inequalities on John domains, Studia Math. 241 (2018), no. 1, 17–39. MR 3732928, DOI 10.4064/sm8488-4-2017
- F. Lopez Garcia, Weighted generalized Korn inequality on John domains, preprint, https://arxiv.org/abs/1612.04449, 2016.
- Stefan Müller, Mathematical problems in thin elastic sheets: scaling limits, packing, crumpling and singularities, Vector-valued partial differential equations and applications, Lecture Notes in Math., vol. 2179, Springer, Cham, 2017, pp. 125–193. MR 3585547
- S. A. Nazarov and A. S. Slutskiĭ, Korn’s inequality for an arbitrary system of thin curved rods, Sibirsk. Mat. Zh. 43 (2002), no. 6, 1319–1331 (Russian, with Russian summary); English transl., Siberian Math. J. 43 (2002), no. 6, 1069–1079. MR 1946232, DOI 10.1023/A:1021121402082
- S. A. Nazarov, Weighted anisotropic Korn’s inequality for a junction of a plate and rods, Mat. Sb. 195 (2004), no. 4, 97–126 (Russian, with Russian summary); English transl., Sb. Math. 195 (2004), no. 3-4, 553–583. MR 2086666, DOI 10.1070/SM2004v195n04ABEH000815
- S. A. Nazarov, Korn’s inequalities for junctions of elastic bodies with thin plates, Sibirsk. Mat. Zh. 46 (2005), no. 4, 876–889 (Russian, with Russian summary); English transl., Siberian Math. J. 46 (2005), no. 4, 695–706. MR 2169404, DOI 10.1007/s11202-005-0070-6
- S. A. Nazarov, Korn’s inequalities for elastic joints of massive bodies, thin plates, and rods, Uspekhi Mat. Nauk 63 (2008), no. 1(379), 37–110 (Russian, with Russian summary); English transl., Russian Math. Surveys 63 (2008), no. 1, 35–107. MR 2406182, DOI 10.1070/RM2008v063n01ABEH004501
- Patrizio Neff, Dirk Pauly, and Karl-Josef Witsch, A canonical extension of Korn’s first inequality to $\mathsf {H}(\textrm {Curl})$ motivated by gradient plasticity with plastic spin, C. R. Math. Acad. Sci. Paris 349 (2011), no. 23-24, 1251–1254 (English, with English and French summaries). MR 2861994, DOI 10.1016/j.crma.2011.10.003
- O. A. Oleĭnik, A. S. Shamaev, and G. A. Yosifian, Mathematical problems in elasticity and homogenization, Studies in Mathematics and its Applications, vol. 26, North-Holland Publishing Co., Amsterdam, 1992. MR 1195131
- L. E. Payne and H. F. Weinberger, On Korn’s inequality, Arch. Rational Mech. Anal. 8 (1961), 89–98. MR 158312, DOI 10.1007/BF00277432
Additional Information
- Davit Harutyunyan
- Affiliation: Department of Mathematics, University of California Santa Barbara, Santa Barbara, California 93106
- MR Author ID: 1072432
- Email: harutyunyan@ucsb.edu
- Hayk Mikayelyan
- Affiliation: Department of Mathematical Sciences, University of Nottingham, Ningbo, 315100 People’s Republic of China
- MR Author ID: 683643
- Email: Hayk.Mikayelyan@nottingham.edu.cn
- Received by editor(s): October 6, 2017
- Received by editor(s) in revised form: October 18, 2018
- Published electronically: May 9, 2019
- Communicated by: Catherine Sulem
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 3635-3647
- MSC (2010): Primary 00A69, 35J65, 74B05, 74B20, 74K25
- DOI: https://doi.org/10.1090/proc/14533
- MathSciNet review: 3981140