Abstract
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 L 2. These estimates are apparently the first to apply with no a priori pointwise hypotheses upon the strain of u. In ℝ3 the integral \(\int\limits_\Omega {}\) e(u) 2 dℋ3 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.
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Kohn, R.V. New integral estimates for deformations in terms of their nonlinear strains. Arch. Rational Mech. Anal. 78, 131–172 (1982). https://doi.org/10.1007/BF00250837
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DOI: https://doi.org/10.1007/BF00250837