On Korn's first inequality with non-constant coefficients

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 Omega subset of R-3 be a bounded Lipschitz domain and let Gamma subset of partial derivativeOmega be a smooth part of the boundary with non-vanishing two-dimensional Lebesgue measure. Define H-o(1,2)(Omega,Gamma) := {phi is an element of H-1,H-2(Omega) \ phi(\Gamma) = 0} and let F-p, F-p(-1) is an element of C-1((&UOmega;) over bar, GL(3,R)) be given with det F-p(x) greater than or equal to mu(+) > 0. Moreover, suppose that Rot F-p is an element of C-1((&UOmega;) over bar, M-3x3). Then There Exists(C+) > 0 For Allphi is an element of H-o(1,2)(Omega,Gamma) : parallel todelphi(.)F(P)(-1)(x) + F-P(-T)(x) (.) delphi(T)parallel to(L2(Omega))(2) greater than or equal to c(+)parallel tophiparallel to(H1,(Omega))(.)(2) Clearly, this result generalizes the classical Korn's first inequality There ExistsC+ > 0 For Allphi is an element of H-o(1,2) (Omega, Gamma) : parallel todelphi + delphi(T)parallel to(L2(Omega))(2) greater than or equal to c(+)parallel tophiparallel to(H1,2(Omega))(2) which is just our result with F-P = II. With slight modifications, we are also able to treat forms of the type parallel toF(P)(x) (.) delphi (.) G(x) + G(x)(T) . delphi(T) . F-P(T)(x)parallel to(p), 1 < p < infinity.