Calculus of Variations and Geometric Measure Theory

D. Spector - S. J. Spector

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

created by spector on 08 Apr 2020



Inserted: 8 apr 2020

Year: 2020

ArXiv: 2004.02368 PDF


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.