Continuity of a Deformation in H¹ as a Function of Its Cauchy-Green Tensor in L¹

Let Ω be a bounded Lipschitz domain in ℝⁿ. The Cauchy-Green, or metric, tensor field associated with a deformation of the set Ω, i.e., a smooth-enough orientation-preserving mapping Θ: Ω → ℝⁿ, is the n × n symmetric matrix field defined by ∇Θ^T (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 H¹(Ω) for the deformations and L¹(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 a hyperelastic material depends on the deformation gradient field ∇Θ through the Cauchy-Green tensor.