A nonlinear Korn inequality with boundary conditions and its relation to the existence of minimizers in nonlinear elasticity

摘要

We establish a nonlinear Korn inequality with boundary conditions showing that the H1-distance between two mappings from Ω∩Rn into Rn preserving orientation is bounded, up to a multiplicative constant, by the L2-distance between their metrics. This inequality is then used to show the existence of a unique minimizer to the total energy of a hyperelastic body, under the assumptions that the Lp-norm of the density of the applied forces is small enough and the stored energy function is bounded from below by a positive definite quadratic function of the Green-Saint Venant strain tensor.