  The uniqueness of equilibrium for a compressible, hyperelastic body subject
to dead-load boundary conditions is considered. It is shown, for both the
displacement and mixed problems, that there cannot be two solutions of the
equilibrium equations of Finite (Nonlinear) Elasticity whose nonlinear strains
are uniformly close to each other. This result is analogous to the result of
Fritz John (Comm. Pure Appl. Math. 25, 617-634, 1972) who proved that, for the
displacement problem, there is a most one equilibrium solution with uniformly
small strains. The proof in this manuscript utilizes Geometric Rigidity; a new
straightforward extension of the Fefferman-Stein inequality to bounded domains;
and, an appropriate adaptation, for Elasticity, of a result from the Calculus
of Variations. Specifically, it is herein shown that the uniform positivity of
the second variation of the energy at an equilibrium solution implies that this
mapping is a local minimizer of the energy among deformations whose gradient is
sufficiently close, in $BMO\cap\, L^1$, to the gradient of the equilibrium
solution.
