Nonlinear Korn inequalities

Let Ω be a bounded and connected open subset of Rⁿ with a Lipschitz-continuous boundary Γ, the set Ω being locally on the same side of Γ, and let Θ : Ω → Rn and Φ : Ω → Rn be two smooth enough "deformations" of the set Ω. Then the classical Korn inequality asserts that, when Θ = id, there exists a constant c such that ‖v‖_H1(Ω) ≤ c(‖v‖_L2(Ω) + ‖∇v +∇v^T‖_L2(Ω)) for all v ∈ H¹ (Ω), where v := (Φ - id) : Ω → Rⁿ denotes the corresponding "displacement" vector field, and where the symmetric tensor field ∇v + ∇v^T : Ω → Sⁿ is nothing but the linear part with respect to v of the difference between the metric tensor fields ∇Φ^T∇Φ and I that respectively correspond to the deformations Φ and Θ = id. Assume now that the identity mapping id is replaced by a more general orientation-preserving immersion Θ ∈ C¹ (Ω;Rⁿ). We then show in particular that, given any 1 < p < ∞ and any q ∈ R such that max{1, p/2} ≤ q ≤ p, there exists a constant C = C (p, q, Θ) such that ‖Φ − Θ‖_W1,p_(Ω) ≤ C ‖Φ − Θ‖Lp_(Ω) + ‖∇Φ^T ∇Φ − ∇Θ^T ∇Θ‖q/p _Lq(Ω)) for all Φ ∈ W^{1, 2q} (Ω) that satisfy det ∇Φ > 0 almost everywhere in Ω. Such an inequality thus constitutes an instance of a "nonlinear Korn inequality", in the sense that the symmetric tensor field ∇Φ^T∇Φ - ∇Θ^T∇Θ : Ω → Sⁿ appearing in its right-hand side is now the exact difference between the metric tensor fields corresponding to the deformations Φ and Θ. We also show that, like in the linear case, an analogous nonlinear Korn inequality holds, but without the norm ‖Φ - Θ‖Lp_(Ω) in its right-hand side, if the difference Φ - Θ vanishes on a subset Γ₀ of Γ with dΓ-meas Γ₀ > 0. The key to providing such nonlinear Korn inequalities is a generalization of the landmark "geometric rigidity lemma in H¹(Ω)" established in 2002 by G. Friesecke, R.D. James, and S. Müller, as later extended to W^1,p (Ω) by S. Conti.