New integral estimates for deformations in terms of their nonlinear strains

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