Continuity of a Deformation in H 1 as a Function of Its Cauchy-Green Tensor in L 1

Let Ω be a bounded Lipschitz domain in R. The CauchyGreen, or metric, tensor field associated with a deformation of the set Ω, i.e., a smooth enough orientation-preserving mapping Θ : Ω → R, is the n× n symmetric matrix field defined by ∇Θ (x)∇Θ(x) at each point x ∈ Ω. We show that, under appropriate assumptions, the deformations depend continuously on their Cauchy-Green tensors, the topologies being those of the spaces H(Ω) for the deformations and L(Ω) for the Cauchy-Green tensors. When n = 3 and Ω is viewed as a reference configuration of an elastic body, this result has potential applications to nonlinear three-dimensional elasticity, since the stored energy function of an hyperelastic material depends on the deformation gradient field ∇Θ through the Cauchy-Green tensor.