×

On the inequalities of Babuška-Aziz, Friedrichs and Horgan-Payne. (English) Zbl 1329.35019

The authors consider the smoothness of a domain \(\Omega\) necessary for several relations appearing in a paper by C. O. Horgan and L. E. Payne [Arch. Ration. Mech. Anal. 82, 165–179 (1983; Zbl 0512.73017)]. First, they investigate the smoothness of \(\Omega\) needed to ensure the relation \(C(\Omega)=1+\Gamma(\Omega)\), where \(C(\Omega)\) is the smallest constant in \(\|u\|^2_{1,\Omega}\leq C\|q\|^2_{0,\Omega}\) for solutions \(u\in H^1_0(\Omega)\) of \(\mathrm{div}(u)=q\). Moreover, \(C(\Omega)=1/\beta(\Omega)^2\) holds for the inf-sup constant \(\beta(\Omega)\) which is important, e.g., for pressure stability in hydromechanics, for proving the Korn inequality in elasticity, for the well-posedness of the Stokes problem and for its finite element approximations and their numerical solution. On the other side, \(\Gamma(\Omega)\) is the best constant in an inequality considered by Friedrichs between conjugate harmonic functions. The relation between this \(\Gamma\) and \(C\) was proved originally for bounded smooth domains in two dimensions, later, by other authors, for Lipschitz and more general domains in any dimension. Here, the present authors show the relation to be true independent on smoothness in dimension 2, further, if one constant is \(\infty\) for certain domains, then the other is \(\infty\), too.
For two-dimensional domains, an upper bound for \(F\) also had been given in the paper by Horgan and Payne for star-shaped domains with respect to a ball. This means an upper bound for the constant \(C\) (and a lower bound for \(\beta\)). Here, the authors give examples of star-shaped domains (with respect to a ball) for which the upper bound by Horgan and Payne is not true but prove that it holds for a number of domains (like regular polygons). They give a new upper bound for all two-dimensional star-shaped domains with respect to a ball. Finally, they mention open questions like the minimal smoothness needed for the relation \(K(\Omega)=2C(\Omega)\) to hold which also appears in the paper of Horgan and Payne [loc. cit.].

MSC:

35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals

Citations:

Zbl 0512.73017
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Acosta G., Durán R.G., Muschietti M.A.: Solutions of the divergence operator on John domains. Adv. Math. 206, 373-401 (2006) · Zbl 1142.35008 · doi:10.1016/j.aim.2005.09.004
[2] Babuška, I., Aziz, A.K.: Survey lectures on the mathematical foundations of the finite element method. In: The mathematical foundations of the finite element method with applications to partial differential equations (Proceedings of symposium on University of Maryland, Baltimore, MD., 1972), Academic Press, New York, pp. 1-359 (1972) · Zbl 0478.76041
[3] Brezzi F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8, 129-151 (1974) · Zbl 0338.90047
[4] Cattabriga L.: Su un problema al contorno relativo al sistema di equazioni di Stokes. Rend. Sem. Mat. Univ. Padova 31, 308-340 (1961) · Zbl 0116.18002
[5] Ciarlet, P.G., Ciarlet, Jr., P.: Another approach to linearized elasticity and a new proof of Korn’s inequality. Math. Models Methods Appl. Sci. 15, 259-271 (2005) · Zbl 1084.74006
[6] Crouzeix, M.: On an operator related to the convergence of Uzawa’s algorithm for the Stokes equation. In: Computational science for the 21st century (Eds. M.-O. Bristeau, G. Etgen, W. Fitzgibbon, J. Lions, J. Périaux and M. Wheeler) Chichester, Wiley, pp. 242-249 (1997) · Zbl 0911.65113
[7] Dauge, M., Bernardi, C., Costabel, M., Girault, V.: On Friedrichs constant and Horgan-Payne angle for LBB condition, In: Twelfth International Conference Zaragoza-Pau on Mathematics, Monogr Mat García Galdeano, Prensas Univ. Zaragoza, vol. 39, pp. 87-100, Zaragoza (2014) · Zbl 1360.35174
[8] Dobrowolski M.: On the LBB condition in the numerical analysis of the Stokes equations. Appl. Numer. Math. 54, 314-323 (2005) · Zbl 1076.65096 · doi:10.1016/j.apnum.2004.09.005
[9] Duran, R., Muschietti, M.-A., Russ, E., Tchamitchian, P.: Divergence operator and Poincaré inequalities on arbitrary bounded domains. Complex Var. Elliptic Equ. 55, 795-816 (2010) · Zbl 1205.35044
[10] Durán, R.G.: An elementary proof of the continuity from \[{L_0^2(\Omega)}\] L02(Ω) to \[{H^1_0(\Omega)^n}\] H01(Ω)n of Bogovskii’s right inverse of the divergence. Revista de la Unión Matemática Argentina53, 59-78 (2012) · Zbl 1278.35189
[11] Friedrichs K.: On certain inequalities and characteristic value problems for analytic functions and for functions of two variables. Trans. Amer. Math. Soc. 41, 321-364 (1937) · JFM 63.0364.01 · doi:10.1090/S0002-9947-1937-1501907-0
[12] Horgan, C.O., Payne, L.E.: On inequalities of Korn, Friedrichs and Babuška-Aziz. Arch. Ration. Mech. Anal. 82, 165-179 (1983) · Zbl 0512.73017
[13] Ladyzhenskaya O., Solonnikov V.: Some problems of vector analysis and generalized formulations of boundary-value problems for the Navier-Stokes equations. J. Sov. Math. 8, 257-286 (1978) · Zbl 0388.35061 · doi:10.1007/BF01566606
[14] Magenes, E., Stampacchia, G.: I problemi al contorno per le equazioni differenziali di tipo ellittico. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze12, 247-358 (1958) · Zbl 0082.09601
[15] Malkus D.S.: Eigenproblems associated with the discrete LBB condition for incompressible finite elements. Int. J. Eng. Sci. 19, 1299-1310 (1981) · Zbl 0457.73051 · doi:10.1016/0020-7225(81)90013-6
[16] Maz’ya, V.: Sobolev spaces with applications to elliptic partial differential equations, vol. 342 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Heidelberg, augmented ed. (2011) · Zbl 1217.46002
[17] Miyazaki Y.: New proofs of the trace theorem of Sobolev spaces. Proc. Jpn Acad. Ser. A Math. Sci. 84, 112-116 (2008) · Zbl 1170.46034 · doi:10.3792/pjaa.84.112
[18] Nečas, J.: Les méthodes directes en théorie des équations elliptiques. Masson-Academia, Paris-Prague, 1967 · Zbl 1225.35003
[19] Oden J.T., Kikuchi N., Song Y.J.: Penalty-finite element methods for the analysis of Stokesian flows. Comput. Methods Appl. Mech. Eng. 31, 297-329 (1982) · Zbl 0478.76041 · doi:10.1016/0045-7825(82)90010-X
[20] Stoyan G.: Towards discrete Velte decompositions and narrow bounds for inf-sup constants. Comput. Math. Appl. 38, 243-261 (1999) · Zbl 0954.65080 · doi:10.1016/S0898-1221(99)00254-0
[21] Stoyan G.: Iterative Stokes solvers in the harmonic Velte subspace. Computing 67, 13-33 (2001) · Zbl 0992.65129 · doi:10.1007/s006070170014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.