Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity

In a previous work, it was shown how the linearized strain tensor field e:=½ (∇u^T+∇u) ∈ L² (Ω) can be considered as the sole unknown in the Neumann problem of linearized elasticity posed over a domain Ω ⊂ R³, instead of the displacement vector field u ∈ H¹ (Ω) 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 u=0 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 e ∈ L² (Ω).