On Korn’s inequalities. (English) Zbl 0698.35067

The authors prove the Korn-type inequality \[ {\mathcal D}(u,\Omega)\leq C_ 1[E(u,\Omega)+{\mathcal D}(u,Q_{R_ 1})], \] where \(E(u,\Omega)=\int_{\Omega}\sum^{n}_{i,j=1}| e_{ij}(u)|^ 2dx\), \(e_{ij}(u)=\partial u_ i/\partial x_ j+\partial u_ j/\partial x_ i\), \({\mathcal D}(u,\Omega)=\int_{\Omega}\sum^{n}_{i,j=1}| \partial u_ i/\partial x_ j|^ 2dx,\) \(x,u\in {\mathbb{R}}^ n\), \(Q_{R_ 1}\) denotes a ball of radius \(R_ 1\), which had its origins in the theory of elasticity.
The authors introduce \((C_ 1-C_{19})\) which depend on the dimension of the space, on \(R_ 1\) and R, which is the diameter of domain \(\Omega\), and prove several inequalities for star-shaped, Lipschitz and other domains.
The importance of Korn inequalities has been amply demonstrated and the inequalities proved in this paper will be important in the study of elliptic equations and in applications to mechanics and elasticity.
Reviewer: V.Komkov


35J99 Elliptic equations and elliptic systems
26D10 Inequalities involving derivatives and differential and integral operators