Korn inequality and divergence operator: Counterexamples and optimality of weighted estimates

Acosta, Gabriel; Durán, Ricardo; López García, Fernando

doi:10.1090/S0002-9939-2012-11408-X

Korn inequality and divergence operator: Counterexamples and optimality of weighted estimates
HTML articles powered by AMS MathViewer

by Gabriel Acosta, Ricardo G. Durán and Fernando López García

Proc. Amer. Math. Soc. 141 (2013), 217-232

DOI: https://doi.org/10.1090/S0002-9939-2012-11408-X

Published electronically: May 22, 2012

PDF | Request permission

Abstract:

The Korn inequality and related results on solutions of the divergence in Sobolev spaces have been widely studied since the pioneering works by Korn and Friedrichs. In particular, it is known that this inequality is valid for Lipschitz domains as well as for the more general class of John domains. On the other hand, a few known counterexamples show that those results are not valid for certain bounded domains having external cusps.

The goal of this paper is to give very simple counterexamples for a class of cuspidal domains in $\mathbb {R}^n$. Moreover, we show that these counterexamples can be used to prove the optimality of recently obtained results involving weighted Sobolev spaces.

References

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
Gabriel Acosta, Ricardo G. Durán, and María A. Muschietti, Solutions of the divergence operator on John domains, Adv. Math. 206 (2006), no. 2, 373–401. MR 2263708, DOI 10.1016/j.aim.2005.09.004
M. E. Bogovskiĭ, Solution of the first boundary value problem for an equation of continuity of an incompressible medium, Dokl. Akad. Nauk SSSR 248 (1979), no. 5, 1037–1040 (Russian). MR 553920
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258, DOI 10.1007/978-1-4757-4338-8
Philippe G. Ciarlet, Introduction to linear shell theory, Series in Applied Mathematics (Paris), vol. 1, Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris; North-Holland, Amsterdam, 1998. MR 1648549
Lars Diening, Michael Růžička, and Katrin Schumacher, A decomposition technique for John domains, Ann. Acad. Sci. Fenn. Math. 35 (2010), no. 1, 87–114. MR 2643399, DOI 10.5186/aasfm.2010.3506
Manfred Dobrowolski, On the LBB condition in the numerical analysis of the Stokes equations, Appl. Numer. Math. 54 (2005), no. 3-4, 314–323. MR 2149355, DOI 10.1016/j.apnum.2004.09.005
R. G. Durán, The inf-sup condition for the Stokes equations: A constructive approach in general domains, Mathematisches Forschungsinstitut Oberwolfach, Workshop on Gemischte und nicht-standard Finite-Elemente-Methoden mit Anwendungen, Extended abstract 5, pp. 270-272, 2005.
Ricardo Duran, Maria-Amelia Muschietti, Emmanuel Russ, and Philippe Tchamitchian, Divergence operator and Poincaré inequalities on arbitrary bounded domains, Complex Var. Elliptic Equ. 55 (2010), no. 8-10, 795–816. MR 2674865, DOI 10.1080/17476931003786659
Ricardo G. Durán and Fernando López García, Solutions of the divergence and analysis of the Stokes equations in planar Hölder-$\alpha$ domains, Math. Models Methods Appl. Sci. 20 (2010), no. 1, 95–120. MR 2606245, DOI 10.1142/S0218202510004167
Ricardo G. Durán and Fernando López García, Solutions of the divergence and Korn inequalities on domains with an external cusp, Ann. Acad. Sci. Fenn. Math. 35 (2010), no. 2, 421–438. MR 2731700, DOI 10.5186/aasfm.2010.3527
Kurt Friedrichs, On certain inequalities and characteristic value problems for analytic functions and for functions of two variables, Trans. Amer. Math. Soc. 41 (1937), no. 3, 321–364. MR 1501907, DOI 10.1090/S0002-9947-1937-1501907-0
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
Giuseppe Geymonat and Gianni Gilardi, Contre-exemples à l’inégalité de Korn et au lemme de Lions dans des domaines irréguliers, Équations aux dérivées partielles et applications, Gauthier-Villars, Éd. Sci. Méd. Elsevier, Paris, 1998, pp. 541–548 (French, with French summary). MR 1648239
C. O. Horgan, Korn’s inequalities and their applications in continuum mechanics, SIAM Rev. 37 (1995), no. 4, 491–511. MR 1368384, DOI 10.1137/1037123
C. O. Horgan and L. E. Payne, On inequalities of Korn, Friedrichs and Babuška-Aziz, Arch. Rational Mech. Anal. 82 (1983), no. 2, 165–179. MR 687553, DOI 10.1007/BF00250935
A. Korn, Die Eigenschwingungen eines elastischen Korpers mit ruhender Oberflache, Akad. der Wissensch Munich, Math-phys. Kl, Beritche, 36, pp. 351-401, 1906.
A. Korn, Ubereinige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen, Bulletin Internationale, Cracovie Akademie Umiejet, Classe de sciences mathématiques et naturelles, pp. 705-724, 1909.
Norbert Weck, Local compactness for linear elasticity in irregular domains, Math. Methods Appl. Sci. 17 (1994), no. 2, 107–113. MR 1258259, DOI 10.1002/mma.1670170204