Korn’s inequality in anisotropic Sobolev spaces

• Autorzy: Benavides Gonzalo A. , Domínguez-Rivera Sebastián A.

Identyfikatory

DOI

10.1515/jaa-2023-0031

• Języki publikacji: EN

Abstrakty

EN

Korn’s inequality has been at the heart of much exciting research since its first appearance in the beginning of the 20th century. Many are the applications of this inequality to the analysis and construction of discretizations of a large variety of problems in continuum mechanics. In this paper, we prove that the classical Korn inequality holds true in anisotropic Sobolev spaces. We also prove that an extension of Korn’s inequality, involving non-linear continuous maps, is valid in such spaces. Finally, we point out that another classical inequality, namely Poincaré’s inequality, also holds in anisotropic Sobolev spaces.

Słowa kluczowe

EN

Korn’s inequality; Poincaré inequality; anisotropic Sobolev spaces; continuum

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2023
• Tom: Vol. 29, nr 2
• Strony: 367--377
• Opis fizyczny: Bibliogr. 59 poz.

Twórcy

autor

Benavides Gonzalo A.

• Department of Mathematics, University of Maryland, College Park, MD 20742, USA

• https://orcid.org/0000-0002-6072-0427

autor

Domínguez-Rivera Sebastián A.

• Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada

• https://orcid.org/0000-0002-3542-8539

Bibliografia

• [1] G. Acosta and R. G. Durán, Divergence Operator and Related Inequalities, Springer Briefs Math., Springer, New York, 2017.
• [2] G. Acosta, R. G. Durán and A. L. Lombardi, Weighted Poincaré and Korn inequalities for Hölder α domains, Math. Methods Appl. Sci. 29 (2006), no. 4, 387-400.
• [3] 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.
• [4] G. Acosta and I. Ojea, Korn’s inequalities for generalized external cusps, Math. Methods Appl. Sci. 39 (2016), no. 17, 4935-4950.
• [5] C. Amrouche and V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J. 44(119) (1994), no. 1, 109-140.
• [6] S. Bauer and D. Pauly, On Korn’s first inequality for mixed tangential and normal boundary conditions on bounded Lipschitz domains in ℝN, Ann. Univ. Ferrara Sez. VII Sci. Mat. 62 (2016), no. 2, 173-188.
• [7] 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.
• [8] O. V. Besov, V. P. Il’in and S. M. Nikol’skiĭ, Integral Representations of Functions and Imbedding Theorems. Vol. I, V. H. Winston & Sons, Washington, 1978.
• [9] O. V. Besov, V. P. Il’in and S. M. Nikol’skiĭ, Integral representations of functions and imbedding theorems. Vol. II, V. H. Winston & Sons, Washington, 1979.
• [10] S. C. Brenner, Korn’s inequalities for piecewise H1 vector fields, Math. Comp. 73 (2004), no. 247, 1067-1087.
• [11] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
• [12] M. Chipot, On inequalities of Korn’s type, J. Math. Pures Appl. (9) 148 (2021), 199-220.
• [13] P. G. Ciarlet, On Korn’s inequality, Chinese Ann. Math. Ser. B 31 (2010), no. 5, 607-618.
• [14] P. G. Ciarlet, Mathematical Elasticity. Volume I. Three-Dimensional Elasticity, Class. Appl. Math. 84, Society for Industrial and Applied Mathematics, Philadelphia, 2022.
• [15] 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), no. 2, 259-271.
• [16] P. G. Ciarlet and C. Mardare, Nonlinear Korn inequalities, J. Math. Pures Appl. (9) 104 (2015), no. 6, 1119-1134.
• [17] S. Conti, D. Faraco and F. Maggi, A new approach to counterexamples to L1 estimates: Korn’s inequality, geometric rigidity, and regularity for gradients of separately convex functions, Arch. Ration. Mech. Anal. 175 (2005), no. 2, 287-300.
• [18] S. Dain, Generalized Korn’s inequality and conformal Killing vectors, Calc. Var. Partial Differential Equations 25 (2006), no. 4, 535-540.
• [19] A. Damlamian, Some remarks on Korn inequalities, Chinese Ann. Math. Ser. B 39 (2018), no. 2, 335-344.
• [20] 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.
• [21] S. Domínguez, Steklov eigenvalues for the Lamé operator in linear elasticity, J. Comput. Appl. Math. 394 (2021), Paper No. 113558.
• [22] S. A. Domínguez-Rivera, N. Nigam and J. S. Ovall, Korn’s inequality and eigenproblems for the Lamé operator, Comput. Methods Appl. Math. 22 (2022), no. 4, 821-837.
• [23] R. G. Durán and M. A. Muschietti, The Korn inequality for Jones domains, Electron. J. Differential Equations 2004 (2004), Paper No. 127.
• [24] G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique, Trav. Rech. Math. 21, Dunod, Paris, 1972.
• [25] V. A. Eremeyev, F. dell’Isola, C. Boutin and D. Steigmann, Linear pantographic sheets: existence and uniqueness of weak solutions, J. Elasticity 132 (2018), no. 2, 175-196.
• [26] P. M. Fitzpatrick, Advanced Calculus, Pure Appl. Undergrad. Texts 5, American Mathematical Society, Providence, 2006.
• [27] M. Friedrich, A piecewise Korn inequality in SBD and applications to embedding and density results, SIAM J. Math. Anal. 50 (2018), no. 4, 3842-3918.
• [28] 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.
• [29] G. N. Gatica, An augmented mixed finite element method for linear elasticity with non-homogeneous Dirichlet conditions, Electron. Trans. Numer. Anal. 26 (2007), 421-438.
• [30] G. N. Gatica, A Simple Introduction to the Mixed Finite Element Method, Springer Briefs Math., Springer, Cham, 2014.
• [31] C. Gavioli and P. Krejčí, On a viscoelastoplastic porous medium problem with nonlinear interaction, SIAM J. Math. Anal. 53 (2021), no. 1, 1191-1213.
• [32] Q. Han, Compact Sobolev embeddings and positive solutions to a quasilinear equation with mixed nonlinearities, J. Math. Anal. Appl. 481 (2020), no. 2, Paper No. 123150.
• [33] Q. Han, Compact Sobolev-Slobodeckij embeddings and positive solutions to fractional Laplacian equations, Adv. Nonlinear Anal. 11 (2022), no. 1, 432-453.
• [34] C. O. Horgan, Korn’s inequalities and their applications in continuum mechanics, SIAM Rev. 37 (1995), no. 4, 491-511.
• [35] L. Hörmander, The Analysis of Linear Partial Differential Operators. I, Class. Math., Springer, Berlin, 2003.
• [36] 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.
• [37] R. Jiang and A. Kauranen, Korn inequality on irregular domains, J. Math. Anal. Appl. 423 (2015), no. 1, 41-59.
• [38] R. Jiang and A. Kauranen, Korn’s inequality and John domains, Calc. Var. Partial Differential Equations 56 (2017), no. 4, Paper No. 109.
• [39] 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.
• [40] A. Korn, Abhandlungen zur Elastizitätstheorie. Die Eigenschwingungen eines elastischen körpers mit ruhender oberfläche, K. B. Akademie der Wissenschaften, München, 1907.
• [41] 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.
• [42] A. Korn, Über einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen, Bull. Int. Acad. Sci. Cracovie 9 (1909), 705-724.
• [43] P. Lewintan, S. Müller and P. Neff, Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy, Calc. Var. Partial Differential Equations 60 (2021), no. 4, Paper No. 150.
• [44] P. Lewintan and P. Neff, Lp-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with p-integrable exterior derivative, C. R. Math. Acad. Sci. Paris 359 (2021), 749-755.
• [45] P. Lewintan and P. Neff, Lp-trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions, Proc. Roy. Soc. Edinburgh Sect. A 152 (2022), no. 6, 1477-1508.
• [46] F. López-García, Weighted generalized Korn inequalities on John domains, Math. Methods Appl. Sci. 41 (2018), no. 17, 8003-8018.
• [47] F. López-García, Weighted Korn inequalities on John domains, Studia Math. 241 (2018), no. 1, 17-39.
• [48] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University, Cambridge, 2000.
• [49] P. Neff, On Korn’s first inequality with non-constant coefficients, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), no. 1, 221-243.
• [50] P. Neff, D. Pauly and K.-J. Witsch, A Korn’s inequality for incompatible tensor fields, PAMM 11 (2011), no. 1, 683-684.
• [51] P. Neff, D. Pauly and K.-J. Witsch, Maxwell meets Korn: A new coercive inequality for tensor fields in ℝN×N with square-integrable exterior derivative, Math. Methods Appl. Sci. 35 (2012), no. 1, 65-71.
• [52] P. Neff and W. Pompe, Counterexamples in the theory of coerciveness for linear elliptic systems related to generalizations of Korn’s second inequality, ZAMM Z. Angew. Math. Mech. 94 (2014), no. 9, 784-790.
• [53] S. M. Nikol’skiĭ, Imbedding, continuation and approximation theorems for differentiable functions of several variables, Uspehi Mat. Nauk 101 (1961), no. 5, 63-114.
• [54] J. A. Nitsche, On Korn’s second inequality, RAIRO Anal. Numér. 15 (1981), no. 3, 237-248.
• [55] D. Ornstein, A non-equality for differential operators in the L1 norm, Arch. Ration. Mech. Anal. 11 (1962), 40-49.
• [56] J. Rákosník, Some remarks to anisotropic Sobolev spaces. I, Beiträge Anal. 13 (1979), 55-68.
• [57] J. Rákosník, Some remarks to anisotropic Sobolev spaces. II, Beiträge Anal. 15 (1980), 127-140.
• [58] T. W. Ting, Generalized Korn’s inequalities, Tensor (N. S.) 25 (1972), 295-302.
• [59] L.-H. Wang, On Korn’s inequality, J. Comput. Math. 21 (2003), no. 3, 321-324.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2024).