Poincaré meets Korn via Maxwell: Extending Korn's first inequality to incompatible tensor fields

Abstract

For a bounded domain Ω⊂R3 with Lipschitz boundary Γ and some relatively open Lipschitz subset Γt≠∅ of Γ, we prove the existence of some c>0, such that(0.1)c‖T‖L2(Ω,R3×3)≤‖symT‖L2(Ω,R3×3)+‖CurlT‖L2(Ω,R3×3) holds for all tensor fields in H(Curl;Ω), i.e., for all square-integrable tensor fields T:Ω→R3×3 with square-integrable generalized rotation CurlT:Ω→R3×3, having vanishing restricted tangential trace on Γt. If Γt=∅, (0.1) still holds at least for simply connected Ω and for all tensor fields T∈H(Curl;Ω) which are L2(Ω)-perpendicular to so(3), i.e., to all skew-symmetric constant tensors. Here, both operations, Curl and tangential trace, are to be understood row-wise.For compatible tensor fields T=∇v, (0.1) reduces to a non-standard variant of the well known Korn's first inequality in R3, namelyc‖∇v‖L2(Ω,R3×3)≤‖sym∇v‖L2(Ω,R3×3) for all vector fields v∈H1(Ω,R3), for which ∇vn, n=1,…,3, are normal at Γt. On the other hand, identifying vector fields v∈H1(Ω,R3) (having the proper boundary conditions) with skew-symmetric tensor fields T, (0.1) turns to Poincaré's inequality since2c‖v‖L2(Ω,R3)=c‖T‖L2(Ω,R3×3)≤‖CurlT‖L2(Ω,R3×3)≤2‖∇v‖L2(Ω,R3). Therefore, (0.1) may be viewed as a natural common generalization of Korn's first and Poincaré's inequality. From another point of view, (0.1) states that one can omit compatibility of the tensor field T at the expense of measuring the deviation from compatibility through CurlT. Decisive tools for this unexpected estimate are the classical Korn's first inequality, Helmholtz decompositions for mixed boundary conditions and the Maxwell estimate.