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In this manuscript two $BMO$ estimates are obtained, one for LinearElasticity 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 aconstant times the $BMO$-seminorm of the symmetric part of its gradient, thatis, a Korn inequality in $BMO$. The uniqueness of equilibrium for a finitedeformation whose principal stresses are everywhere nonnegative is thenconsidered. It is shown that when the second variation of the energy, whenconsidered as a function of the strain, is uniformly positive definite at suchan equilibrium solution, then there is a $BMO$-neighborhood in strain spacewhere there are no other equilibrium solutions.

BMO and Elasticity: Korn's Inequality; Local Uniqueness in Tension