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BMO and Elasticity: Korn’s Inequality; Local Uniqueness in Tension
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AbstractIn 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.
Springer Science and Business Media LLC
Title: BMO and Elasticity: Korn’s Inequality; Local Uniqueness in Tension
Description:
AbstractIn 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.
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