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

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.

Fractional Korn and Hardy-type inequalities for vector fields in half space

We prove a fractional Hardy-type inequality for vector fields over the half space based on a modified fractional semi-norm. A priori, the modified semi-norm is not known to be equivalent to the standard fractional semi-norm and in fact gives a smaller norm, in general. As such, the inequality we prove improves the classical fractional Hardy inequality for vector fields. We will use the inequality to establish the equivalence of a space of functions (of interest) defined over the half space with the classical fractional Sobolev spaces, which amounts to prove a fractional version of the classical Korn’s inequality.

Analogues of Korn’s inequality on Heisenberg groups

Two analogues of Korn’s inequality on Heisenberg groups are constructed. First, the norm of the horizontal differential is estimated in terms of its symmetric part. Second, Korn’s inequality is treated as a coercive estimate for a differential operator whose kernel coincides with the Lie algebra of the isometry group. For this purpose, we construct a differential operator whose kernel coincides with the Lie algebra of the isometry group on Heisenberg groups and prove a coercive estimate for this operator. Additionally, a coercive estimate is proved for a differential operator whose kernel coincides with the Lie algebra of the group of conformal mappings on Heisenberg groups.

On a decomposition of regular domains into John domains with uniform constants

We derive a decomposition result for regular, two-dimensional domains into John domains with uniform constants. We prove that for every simply connected domain Ω ⊂ ℝ2 with C1-boundary there is a corresponding partition Ω = Ω1 ⋃ … ⋃ ΩN with Σj=1NH1(∂Ωj\∂Ω)≤θ such that each component is a John domain with a John constant only depending on θ. The result implies that many inequalities in Sobolev spaces such as Poincaré’s or Korn’s inequality hold on the partition of Ω for uniform constants, which are independent of Ω.