New integral estimates for deformations in terms of their nonlinear strains

摘要: If u is a bi-Lipschitzian deformation of bounded Lipschitz domain Ω in ln (n≧2), we show that the LP norm (p≧1, p≠n) certain “nonlinear strain function” e(u) associated with dominates distance Lq (q= np/(n−p) if p n) from to suitably chosen rigid motion ℝn. This work extends F. John, who proved corresponding estimates for p}>1 under hypothesis has “uniformly small strain”. We also obtain bound oscillation Du L2. These are apparently first apply no priori pointwise hypotheses upon u. In ℝ3 integral \(\int\limits_\Omega {}\)e(u)2dℋ3 dominated by typical hyperelastic energy functionals proposed literature modeling behavior rubber; thus case n=3, p=2 gives general deformations such materials terms nonlinear elastic work.