Nonlinear Korn inequalities

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Original languageEnglish
Pages (from-to)1119-1134
Journal / PublicationJournal des Mathematiques Pures et Appliquees
Issue number6
Online published7 Jul 2015
Publication statusPublished - Dec 2015


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 ‖vH1(Ω) c(‖vL2(Ω) + ‖+vTL2(Ω)) for all vH(Ω), where := (Φ - id) : Ω → Rn denotes the corresponding "displacement" vector field, and where the symmetric tensor field v + v: Ω → Sn is nothing but the linear part with respect to v of the difference between the metric tensor fields ∇ΦT∇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 ∈ C(Ω;Rn). We then show in particular that, given any 1 < p < ∞ and any q ∈ R such that max{1, p/2} ≤ qp, there exists a constant C = (p, q, Θ) such that ‖ΦΘW1,p(Ω)CΦΘLp(Ω) + ‖∇ΦT ∇Φ∇ΘT ∇Θq/p Lq(Ω)) for all ΦW1, 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 ∇ΦT∇Φ ∇ΘT∇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 ΘLp(Ω) in its right-hand side, if the difference Φ - Θ vanishes on a subset Γ0 of Γ with dΓ-meas Γ> 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 W1,(Ω) by S. Conti.

Research Area(s)

  • Linear Korn inequalities, Metric tensor, Nonlinear Korn inequalities