New integral estimates for deformations in terms of their nonlinear strains

If u is a bi-Lipschitzian deformation of a bounded Lipschitz domain in ⁿ (n2), we show that the L ^P norm (p1, pn) of a certain nonlinear strain function e(u) associated with u dominates the distance in L ^q (q= np/(n–p) if p if p>n) from u to a suitably chosen rigid motion of ⁿ . 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 L ². These estimates are apparently the first to apply with no a priori pointwise hypotheses upon the strain of u. In ³ the integral e(u) ² d³ 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.