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On the dependence of the constant in Korn’s inequality on parameters characterizing the geometry of the region. (English. Russian original) Zbl 0727.73023

Russ. Math. Surv. 44, No. 6, 187-189 (1989); translation from Usp. Mat. Nauk 44, No. 6(270), 157-158 (1989).
Summary: In this paper we give a new elementary proof of Korn’s inequality for starlike regions, we explain the dependence of the constants in this inequality on the geometric properties of the region, and we prove that these constants are in a certain sense best possible. This enables us to establish Korn’s inequality for many classes of regions of importance in applied problems and which depend on parameters (such as lattices, nets, towers, cranes, linkage, frames, and so on) and to explain the nature of the dependence of their constants on the parametes. In particular, the results of this paper strengthen a number of theorems proved e.g. by the authors [C.R. Acad. Sci., Paris, Sér. I 308, No.16, 483-487 (1989; Zbl 0698.35067)]. There is an extensive literature devoted to Korn’s inequalities and their applications [see the authors, Usp. Mat. Nauk 33, No.5(263), 55-98 (1988; Zbl 0661.73001) and the literature cited therein].

MSC:

74B20 Nonlinear elasticity
26D10 Inequalities involving derivatives and differential and integral operators
35B45 A priori estimates in context of PDEs
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