Some estimates for the oscillation of the deformation gradient (Q700936)

The author generalizes the estimate of deformation gradient \(D \vec \phi \) due to \textit{R. V. Kohn} [Arch. Ration. Mech. Anal. 78, 131-172 (1982; Zbl 0491.73023)] to the case when \(p \neq 2\). The norm \(\parallel D \vec \phi - R \parallel _{L^p(\Omega)}\) is estimated by means of scalar measure \(e(\vec \phi)\) of nonlinear strain (here \(R\) is the gradient of a mapping \(\vec \gamma \) corresponding to some rigid motion, \(\vec \phi : \Omega \to\mathbb{R}^n\) is the deformation.) First, the estimate is given for a deformation \(\vec \phi \in W^{1,p}(\Omega)\) satisfying the condition \(\vec \phi |_{\partial \Omega } = id\). After this, the estimate is derived for general bi-Lipschitzian mapping \(\vec \phi \).