BMO and Elasticity: Korn's Inequality; Local Uniqueness in Tension
arXiv:2004.02368
Abstract
In this manuscript two estimates are obtained, one for Linear Elasticity and one for Nonlinear Elasticity. It is first shown that the -seminorm of the gradient of a vector-valued mapping is bounded above by a constant times the -seminorm of the symmetric part of its gradient, that is, a Korn inequality in . 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 -neighborhood in strain space where there are no other equilibrium solutions.
23 pages