  In this manuscript two $BMO$ estimates are obtained, one for Linear
Elasticity and one for Nonlinear Elasticity. It is first shown that the
$BMO$-seminorm of the gradient of a vector-valued mapping is bounded above by a
constant times the $BMO$-seminorm of the symmetric part of its gradient, that
is, a Korn inequality in $BMO$. The uniqueness of equilibrium for a finite
deformation whose principal stresses are everywhere nonnegative is then
considered. It is shown that when the second variation of the energy, when
considered as a function of the strain, is uniformly positive definite at such
an equilibrium solution, then there is a $BMO$-neighborhood in strain space
where there are no other equilibrium solutions.
