Inégalités de Korn non linéaires dans [formula omitted], avec ou sans conditions aux limites

Abstract

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 ‖Φ−Θ‖W1,p(Ω) can be bounded above in terms of the norm ‖∇ΦT∇Φ−∇ΘT∇Θ‖Lq(Ω) for any q∈R such that max⁡{1,p2}≤q≤p, where (∇ΦT∇Φ−∇ΘT∇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 ∇ΦT∇Φ−I is reduced to its linear part ∇vT+∇v with respect to the vector field v:=Φ−id:Ω¯→Rn.