TY - JOUR

T1 - BMO and Elasticity

T2 - Korn’s Inequality; Local Uniqueness in Tension

AU - Spector, Daniel E.

AU - Spector, Scott J.

N1 - Publisher Copyright:
© 2020, The Author(s).

PY - 2021/1

Y1 - 2021/1

N2 - 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.

AB - 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.

KW - BMO local minimizers

KW - Bounded mean oscillation

KW - Equilibrium solutions

KW - Finite elasticity

KW - Korn’s inequality

KW - Nonlinear elasticity

KW - Small strains

KW - Uniqueness

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U2 - 10.1007/s10659-020-09805-5

DO - 10.1007/s10659-020-09805-5

M3 - Article

AN - SCOPUS:85099032764

SN - 0374-3535

VL - 143

SP - 85

EP - 109

JO - Journal of Elasticity

JF - Journal of Elasticity

IS - 1

ER -