Existence of minimizers for the pure displacement problem in nonlinear elasticity

We show that the total energy of the pure displacement problem in nonlinear elasticity possesses a unique global minimizer for a large class of hyperelastic materials, including that of Saint Venant - Kirchhoff, provided the density of the applied forces are small in L(P)-norm. We also establish a nonlinear Korn inequality with boundary showing that the H(1)-distance between two deformation fields is bounded, up to a multiplicative constant, by the L(2)-distance between their Cauchy-Green strain tensors.