L p -versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with p-integrable exterior derivative

Peter Lewintan,Patrizio Neff

doi:10.5802/crmath.216

Abstract

For $n\ge2$ and $1<p<\infty$ we prove an $L^p$-version of the generalized Korn-type inequality for incompatible, $p$-integrable tensor fields $P:\Omega \to \mathbb{R}^{n\times n}$ having $p$-integrable generalized $\underline{\operatorname{Curl}}$ and generalized vanishing tangential trace $P\,\tau_l=0$ on $\partial \Omega$, denoting by $\{\tau_l\}_{l=1,\ldots, n-1}$ a moving tangent frame on $\partial\Omega$, more precisely we have: $$\| P \|_{L^p(\Omega,\mathbb{R}^{n\times n})}\leq c\,(\| \operatorname{sym} P\|_{L^p(\Omega,\mathbb{R}^{n \times n})} + \|\underline{\operatorname{Curl}} P \|_{L^p(\Omega,(\mathfrak{so}(n))^n)} ),$$ where the generalized $\underline{\operatorname{Curl}}$ is given by $ (\underline{\operatorname{Curl}})_{ijk} :=\partial_i P_{kj}-\partial_j P_{ki}$ and $c=c(n,p,\Omega)>0$.

Highlights

That for a bounded Lipschitz domain Ω ⊂ Rn, the Lions lemma states that f ∈ Lp (Ω) if and only if f ∈ W −1, p (Ω) and ∇ f ∈ W −1, p (Ω, Rn), which is equivalently expressed by the Necas estimate f Lp (Ω) ≤ c f W −1, p (Ω) + ∇ f W −1, p (Ω, Rn )
[1,2,3,4,5] focus on the compatible case, i.e. P = Du, where we deal with general square matrices P ∈ Rn×n, the incompatible case
: P T ek ×ν,Q ∂Ω = Ω curl P T ek,Q Rn×n + 2 P T ek, Div skewQ Rn dx having denoted by Q ∈ W 1, p (Ω, Rn×n) any extension of Q in Ω, where, 〈·, ·〉∂Ω indicates the duality pairing between

Summary

Introduction

Holds for all tensor fields P ∈ W01, p (Curl; Ω, R3×3), i.e., for all P ∈ W 1, p (Curl; Ω, R3×3) with vanishing tangential trace P × ν = 0 (⇔ P τl = 0) on ∂Ω where ν denotes the outward unit normal vector field and {τl }l=1,2,3 a moving tangent frame on ∂Ω and Ω ⊂ R3 is a bounded Lipschitz domain. We extend our results from [6] to the n-dimensional case, generalizing the main result from [8] to the Lp -setting We follow the argumentation scheme presented in [6] closely, emphasizing only the necessary modifications coming from the generalization of the vector product. The latter provides an adequate generalization of the Curl-operator to the n-dimensional setting. The generalized curl of vector fields can be seen as their exterior derivative, see the discussion in [8]

Notations

Function spaces

Main results

Methods