On Korn's First Inequality in a Hardy-Sobolev Space

被引:0
|
作者
Spector, Daniel E. [1 ]
Spector, Scott J. [2 ]
机构
[1] Natl Taiwan Normal Univ, Dept Math, 88,Sect 4,Tingzhou Rd, Taipei 116, Taiwan
[2] Southern Illinois Univ, Dept Math, Carbondale, IL 62901 USA
关键词
Korn's inequality; Hardy-Sobolev spaces; Riesz transforms; Linear elasticity; GEOMETRIC RIGIDITY; SMOOTH DOMAIN; OPERATORS;
D O I
10.1007/s10659-022-09976-3
中图分类号
T [工业技术];
学科分类号
08 ;
摘要
Korn's first inequality states that there exists a constant such that the L-2-norm of the infinitesimal displacement gradient is bounded above by this constant times the L-2-norm of the infinitesimal strain, i.e., the symmetric part of the gradient, for all infinitesimal displacements that are equal to zero on the boundary of a body B. This inequality is known to hold when the L-2-norm is replaced by the Lp-norm for any p is an element of (1, infinity). However, if p = 1 or p = infinity the resulting inequality is false. It was previously shown that if one replaces the L-infinity- norm by the BMO-seminorm (Bounded Mean Oscillation) then one maintains Korn's inequality. (Recall that L-infinity(B) subset of BMO(B) subset of L-p(B) subset of L-1(B), 1 < p < infinity.) In this manuscript it is shown that Korn's inequality is also maintained if one replaces the L-1-norm by the norm in the Hardy space H-1, the predual of BMO. One caveat: the results herein are only applicable to the pure-displacement problem with the displacement equal to zero on the entire boundary of B.
引用
收藏
页码:187 / 198
页数:12
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