Nonlinear Korn inequalities

International audienceLet 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 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. Muller, as later extended to W1,p(Ω) by S. Conti