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International audienceWe establish a nonlinear Korn inequality with boundary conditions showing that the H^1-distance between two mappings from Ω ⊂ R^n into R^n preserving orientation is bounded, up to a multiplicative constant, by the L^2-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 L^p-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

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