Abstract
In this paper, we show that the so-called Korn inequality holds for vector fields with a zero normal or tangential trace on a subset (of positive measure) of the boundary of Lipschitz domains. We further show that the validity of this inequality depends on the geometry of this subset of the boundary. We then consider three eigenvalue problems for the Lamé operator: we constrain the traction in the tangential direction and the normal component of the displacement, the related problem of constraining the normal component of the traction and the tangential component of the displacement, and a third eigenproblem that considers mixed boundary conditions. We show that eigenpairs for these eigenproblems exist on a broad variety of domains. Analytic solutions for some of these eigenproblems are given on simple domains.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-2012285
Funding statement: Sebastián Domínguez thanks the support of the Pacific Institute for the Mathematical Sciences through a PIMS postdoctoral fellowship. Nilima Nigam thanks the support of NSERC through the Discovery program of Canada. The research of Jeffrey Ovall is partially supported by NSF grant DMS-2012285.
Acknowledgements
We are grateful for the suggestions of the anonymous reviewers, which not only improved the readability of the manuscript, but also enabled us to provide additional context for our work (cf. [16]). The authors thank Michael Levitin (University of Reading, UK) who pointed out to us the notion of tangential-normal eigenpairs for the Lamé operator.
References
[1] G. Acosta, R. G. Durán and M. A. Muschietti, Solutions of the divergence operator on John domains, Adv. Math. 206 (2006), no. 2, 373–401. 10.1016/j.aim.2005.09.004Search in Google Scholar
[2] G. Acosta and I. Ojea, Korn’s inequalities for generalized external cusps, Math. Methods Appl. Sci. 39 (2016), no. 17, 4935–4950. 10.1002/mma.3170Search in Google Scholar
[3]
S. Bauer and D. Pauly,
On Korn’s first inequality for mixed tangential and normal boundary conditions on bounded Lipschitz domains in
[4] S. Bauer and D. Pauly, On Korn’s first inequality for tangential or normal boundary conditions with explicit constants, Math. Methods Appl. Sci. 39 (2016), no. 18, 5695–5704. 10.1002/mma.3954Search in Google Scholar
[5] E. Cáceres, J. Guzmán and M. Olshanskii, New stability estimates for an unfitted finite element method for two-phase Stokes problem, SIAM J. Numer. Anal. 58 (2020), no. 4, 2165–2192. 10.1137/19M1266897Search in Google Scholar
[6] M. Chipot, On inequalities of Korn’s type, J. Math. Pures Appl. (9) 148 (2021), 199–220. 10.1016/j.matpur.2020.08.012Search in Google Scholar
[7] P. G. Ciarlet, On Korn’s inequality, Chin. Ann. Math. Ser. B 31 (2010), no. 5, 607–618. 10.1007/s11401-010-0606-3Search in Google Scholar
[8]
S. Conti, D. Faraco and F. Maggi,
A new approach to counterexamples to
[9] A. Damlamian, Some remarks on Korn inequalities, Chin. Ann. Math. Ser. B 39 (2018), no. 2, 335–344. 10.1007/s11401-018-1067-3Search in Google Scholar
[10] F. Demengel and G. Demengel, Functional Spaces for the Theory Of Elliptic Partial Differential Equations, Universitext, Springer, London, 2012. 10.1007/978-1-4471-2807-6Search in Google Scholar
[11] L. Desvillettes and C. Villani, On a variant of Korn’s inequality arising in statistical mechanics, ESAIM Control Optim. Calc. Var. 8 (2002), 603–619. 10.1051/cocv:2002036Search in Google Scholar
[12] S. Domínguez, Steklov eigenvalues for the Lamé operator in linear elasticity, J. Comput. Appl. Math. 394 (2021), Article ID 113558. 10.1016/j.cam.2021.113558Search in Google Scholar
[13] R. G. Durán and M. A. Muschietti, The Korn inequality for Jones domains, Electron. J. Differential Equations 2004 (2004), Paper No. 127. Search in Google Scholar
[14] G. N. Gatica, A Simple Introduction to the Mixed Finite Element Method, Springer Briefs Math., Springer, Cham, 2014. 10.1007/978-3-319-03695-3Search in Google Scholar
[15] C. Gavioli and P. Krejčí, On a viscoelastoplastic porous medium problem with nonlinear interaction, SIAM J. Math. Anal. 53 (2021), no. 1, 1191–1213. 10.1137/20M1340617Search in Google Scholar
[16] I. Hlaváček and J. Nečas, On inequalities of Korn’s type. II. Applications to linear elasticity, Arch. Ration. Mech. Anal. 36 (1970), 312–334. 10.1007/BF00249519Search in Google Scholar
[17] C. O. Horgan, Korn’s inequalities and their applications in continuum mechanics, SIAM Rev. 37 (1995), no. 4, 491–511. 10.1137/1037123Search in Google Scholar
[18] G. C. Hsiao, R. E. Kleinman and G. F. Roach, Weak solutions of fluid-solid interaction problems, Math. Nachr. 218 (2000), 139–163. 10.1002/1522-2616(200010)218:1<139::AID-MANA139>3.0.CO;2-SSearch in Google Scholar
[19] G. C. Hsiao and N. Nigam, A transmission problem for fluid-structure interaction in the exterior of a thin domain, Adv. Differential Equations 8 (2003), no. 11, 1281–1318. 10.57262/ade/1355926118Search in Google Scholar
[20] G. C. Hsiao and T. Sánchez-Vizuet, Time-domain boundary integral methods in linear thermoelasticity, SIAM J. Math. Anal. 52 (2020), no. 3, 2463–2490. 10.1137/19M1298652Search in Google Scholar
[21] G. C. Hsiao, T. Sánchez-Vizuet and F.-J. Sayas, Boundary and coupled boundary–finite element methods for transient wave–structure interaction, IMA J. Numer. Anal. 37 (2017), no. 1, 237–265. 10.1093/imanum/drw009Search in Google Scholar
[22] H. Igel, Computational Seismology: A Practical Introduction, Oxford University, Oxford, 2016. 10.1093/acprof:oso/9780198717409.001.0001Search in Google Scholar
[23] R. Jiang and A. Kauranen, Korn inequality on irregular domains, J. Math. Anal. Appl. 423 (2015), no. 1, 41–59. 10.1016/j.jmaa.2014.09.076Search in Google Scholar
[24] R. Jiang and A. Kauranen, Korn’s inequality and John domains, Calc. Var. Partial Differential Equations 56 (2017), no. 4, Paper No. 109. 10.1007/s00526-017-1196-7Search in Google Scholar
[25] D. S. Jones, Low-frequency scattering by a body in lubricated contact, Quart. J. Mech. Appl. Math. 36 (1983), no. 1, 111–138. 10.1093/qjmam/36.1.111Search in Google Scholar
[26] D. S. Jones, A uniqueness theorem in elastodynamics, Quart. J. Mech. Appl. Math. 37 (1984), no. 1, 121–142. 10.1093/qjmam/37.1.121Search in Google Scholar
[27] P. Kaplický and J. Tichý, Boundary regularity of flows under perfect slip boundary conditions, Cent. Eur. J. Math. 11 (2013), no. 7, 1243–1263. 10.2478/s11533-013-0232-xSearch in Google Scholar
[28] A. Korn, Die Eigenschwingungen eines elastichen Körpers mit ruhender Oberfläche, Akad. Wiss Munich Math-Phys. Kl. Berichte 36 (1906), 351–401. Search in Google Scholar
[29] A. Korn, Solution générale du problème d’équilibre dans la théorie de l’élasticité dans le cas où les efforts sont donnés à la surface, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (2) 10 (1908), 473–473. 10.5802/afst.255Search in Google Scholar
[30] A. Korn, Über einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen, Bull. Int. Acad. Sci. de Cracovie 9 (1909), 705–724. Search in Google Scholar
[31] C. Le Roux, Existence and uniqueness of the flow of second-grade fluids with slip boundary conditions, Arch. Ration. Mech. Anal. 148 (1999), no. 4, 309–356. 10.1007/s002050050164Search in Google Scholar
[32] C. Le Roux, On flows of viscoelastic fluids of Oldroyd type with wall slip, J. Math. Fluid Mech. 16 (2014), no. 2, 335–350. 10.1007/s00021-013-0159-9Search in Google Scholar
[33] C. Le Roux, Flows of incompressible viscous liquids with anisotropic wall slip, J. Math. Anal. Appl. 465 (2018), no. 2, 723–730. 10.1016/j.jmaa.2018.05.020Search in Google Scholar
[34] M. Levitin, P. Monk and V. Selgas, Impedance eigenvalues in linear elasticity, SIAM J. Appl. Math. 81 (2021), no. 6, 2433–2456. 10.1137/21M1412955Search in Google Scholar
[35] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University, Cambridge, 2000. Search in Google Scholar
[36] S. Meddahi, D. Mora and R. Rodríguez, Finite element spectral analysis for the mixed formulation of the elasticity equations, SIAM J. Numer. Anal. 51 (2013), no. 2, 1041–1063. 10.1137/120863010Search in Google Scholar
[37] D. Natroshvili, G. Sadunishvili and I. Sigua, Some remarks concerning Jones eigenfrequencies and Jones modes, Georgian Math. J. 12 (2005), no. 2, 337–348. 10.1515/GMJ.2005.337Search in Google Scholar
[38] J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer Monogr. Math., Springer, Heidelberg, 2011. 10.1007/978-3-642-10455-8Search in Google Scholar
[39] J. A. Nitsche, On Korn’s second inequality, RAIRO Anal. Numér. 15 (1981), no. 3, 237–248. 10.1051/m2an/1981150302371Search in Google Scholar
[40] R. W. Ogden, Nonlinear Elastic Deformations, Dover, New York, 1997. Search in Google Scholar
[41]
D. Ornstein,
A non-equality for differential operators in the
[42] T. Yin, G. C. Hsiao and L. Xu, Boundary integral equation methods for the two-dimensional fluid-solid interaction problem, SIAM J. Numer. Anal. 55 (2017), no. 5, 2361–2393. 10.1137/16M1075673Search in Google Scholar
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