On Korn's first inequality with non-constant coefficients

Source abstract

In this paper we prove a Korn-type inequality with non-constant coefficients which arises from applications in elasto-plasticity at large deformations. More precisely, let Ω ⊂ R 3 be a bounded Lipschitz domain and let Γ ⊂ ∂Ω be a smooth part of the boundary with non-vanishing two-dimensional Lebesgue measure. Define and let be given with det F p ( x ) ≥ μ + > 0. Moreover, suppose that Rot . Then Clearly, this result generalizes the classical Korn's first inequality which is just our result with F p = 11. With slight modifications, we are also able to treat forms of the type