Expression of Dirichlet boundary conditions in terms of the Cauchy-Green tensor field

In a previous work, it was shown how the Cauchy-Green tensor field C := ∇Φ^T∇Φ ∈ W^2,s (Ω; S³_>), s >3/2, can be considered as the sole unknown in the homogeneous Dirichlet problem of nonlinear elasticity posed over a domain Ω ⊂ R³, instead of the deformation Φ ∈ W^3,s (Ω; R³) in the usual approach. The purpose of this Note is to show that the same approach applies as well to the Dirichlet-Neumann problem. To this end, we show how the boundary condition Φ = Φ₀ on a portion Γ₀ of the boundary of Ω can be recast, again as boundary conditions on Γ₀, but this time expressed only in terms of the new unknown C ∈ W^2,s (Ω; S³_>).