Une inégalité de Korn non linéaire dans W^2,p, p>n

Let Ω be a bounded and connected open subset of Rn that satisfies the uniform interior cone property and let p>n. We establish a nonlinear Korn inequality in W^2,p, which provides an upper bound of the ‖{dot operator}‖W2,p(Ω)-norm of the difference between two immersions ϕ∈W^2,p(Ω) and ϕ~∈W2,p(Ω) in terms of the ‖{dot operator}‖W1,p(Ω)-norm of the difference between their associated metric tensors ∇ϕT∇ϕ∈W1,p(Ω) and ∇ϕ~T∇ϕ~∈W1,p(Ω).Second, let Ω be a bounded, simply-connected, open subset of Rn with a Lipschitz boundary, the set Ω being locally on the same side of its boundary. Using the above nonlinear Korn inequality in W^2,p, we establish the local Lipschitz-continuity of the mapping C∈W1,p(Ω)→ϕ∈W2,p(Ω), which is well-defined when the components of the Riemann curvature tensor associated with C vanish in D'(Ω), the immersion ϕ∈W^2,p(Ω) being the solution, unique up to an isometry of En, of the equation ∇ϕ^T∇ϕ=C in Ω.