Inégalités de Korn non linéaires dans Rn, avec ou sans conditions aux limites

Let Ω be a bounded open subset of Rⁿ with a Lipschitz boundary. Given two smooth enough immersions Φ : Ω → Rⁿ et Θ : Ω → Rⁿ with the same orientation, we establish various nonlinear Korn inequalities that show that, for any 1 < p < ∞, the norm ǁΦ − Θǁ_W1,p _(Ω) can be bounded above in terms of the norm ǁ∇^T∇Φ − ∇Θ^T∇Θǁ_Lq _(Ω) for any q ∈ R such that max{1, p/2 } ≤ q ≤ p, where (∇Θ^T∇Φ - ∇Θ^T∇Θ) thus represents the exact difference between the metrics corresponding to the immersions Φ and Θ. Such inequalities generalize the well-known linear Korn inequalities, where, when Θ = id, the exact difference ∇Φ^T∇Φ - I is reduced to its linear part ∇vT + ∇v with respect to the vector field v := Φ - id : Ω → Rⁿ.