Abstract
We are concerned with the coerciveness of the strain energy E(u) (in linear elasticity) associated with a displacement vector u on the Sobolev space H1 (Ω) or its subspaces, a domain Ω in ℝn representing an isotropic elastic body—certain specific cases are called “Korn's inequalities”. Sufficient (and necessary) conditions on the “Lamé moduli” for E(·) to be coercive (or uniformly positive) on such spaces are given, and the associated best possible constants are obtained for some cases.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. Bernstein and R.A. Toupin, Korn inequalities for the sphere and circle. Arch. Rational Mech. Anal. 6 (1960) 51–64.
C.M. Dafermos, Some remarks on Korn's inequality. Z. Angew. Math. Phys. 19 (1968) 913–920.
G. Duvaut and J.L. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972).
G. Fichera, Existence theorems in elasticity. In: S. Flügge and C. Truesdell (eds.), Handbuch der Physik, Pd. 6a/2. Springer, Berlin (1965) 347–389.
K.O. Friedrichs, On the boundary-value problems of the theory of elasticity and Korn's inequality. Ann. of Math. 48 (1947) 441–471.
J. Gobert, Une inégalité fondamentale de la théorie de l'élasticité. Bull. Soc. Roy. Sci. Liège 31 (1962) 182–191.
C.O. Horgan, On Korn's inequality for incompressible media. SIAM J. Appl. Math. 28. (1975) 419–430.
C.O. Horgan, Inequalities of Korn and Friedrichs in elasticity and potential theory. Z. Angew. Math. Phys. 26 (1975) 155–164.
C.O. Horgan and J.K. Knowles, Eigenvalue problems associated with Korn's inequalities. Arch. Rational Mech. Anal. 40 (1971) 384–402.
C.O. Horgan and L.E. Payne, On inequalities of Korn, Friedrichs and Babuška-Aziz. Arch. Rational Mech. Anal. 82 (1983) 165–179.
A. Korn, Die Eigenschwingungen eines elastische Köpers mit ruhender Oberfläche. Akad. Wiss., Munchen, Math. Phys. Kl. 36 (1906) 351–402.
A. Korn, Über einige Ungleischungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen. Akad. Umiejet. Krakow Bulletin Int. (1909) 705–724.
S.G. Mikhlin, The problem of the minimum of a quadratic functional. Holden-Day, San Francisco-London (1965).
S.G. Mikhlin, The spectrum of a family of operators in the theory of elasticity. Russian Math. Surveys 28–3 (1973) 45–88.
J.A. Nitsche, On Korn's second inequality. RAIRO J. Numer. Anal. 15 (1981) 237–248.
L.E. Payne and H.F. Weinberger, On Korn's inequality. Arch. Rational Mech. Anal. 8 (1961) 89–98.
Y. Shibata and H. Soga, Scattering theory for the elastic wave equation. Publ. RIMS, Kyoto Univ. 25 (1989) 861–887.
H.C. Simpson and S.J. Spector, On the positivity of the second variation in finite elasticity. Arch. Rational Mech. Anal. 98 (1987) 1–30.
M.J. Strauss, Variations of Korn's and Sobolev's inequalities. In: D.C. Spencer (ed.) Partial Differential Equations (Proc. Simposia in Pure Mathematics. Vol. 23) AMS Providence (1973) 207–214.
T.W. Ting, Generalized Korn's inequalities. Tensor 25 (1972) 295–302.