$L^p$-trace-free generalized Korn inequalities for incompatible tensor fields in three space dimensions

Abstract

For $1<p<\infty$ we prove an $L^p$-version of the generalized trace-free Korn inequality for incompatible tensor fields $P$ in $W_0^{1,p}(\mathrm{Curl};\Omega ,{\mathbb{R}}^{3\times 3})$. More precisely, let $\Omega \subset {\mathbb{R}}^3$ be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that \[ \|{ P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}\leq c\,\left(\|{\operatorname{dev} \operatorname{sym} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})} + \|{ \operatorname{dev} \operatorname{Curl} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}\right) \] holds for all tensor fields $P\in W_0^{1,p}(\mathrm{Curl};\Omega ,{\mathbb{R}}^{3\times 3})$, i.e., for all $P\in W^{1,p}(\mathrm{Curl};\Omega ,{\mathbb{R}}^{3\times 3})$ with vanishing tangential trace $P\times \nu =0$ on $\partial \Omega$ where $\nu$ denotes the outward unit normal vector field to $\partial \Omega$ and $\mathrm{dev}P:=P-\frac{1}{3}\mathrm{tr}(P)1_3$ denotes the deviatoric (trace-free) part of $P$. We also show the norm equivalence \[ \|{ P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}+\|{\operatorname{Curl} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}\leq c\,\left(\|{\operatorname{dev} \operatorname{sym} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})} + \|{ \operatorname{dev}\operatorname{Curl} P }\|_{L^p(Ω,\mathbb{R}^{3\times3})}\right) \] for tensor fields $P\in W_0^{1,p}(\mathrm{Curl};\Omega ,{\mathbb{R}}^{3\times 3})$. These estimates also hold true for tensor fields with vanishing tangential trace only on a relatively open (non-empty) subset $\Gamma \subseteq \partial \Omega$ of the boundary.

References in corpus (8)

• A unifying perspective: the relaxed linear micromorphic continuum
• Generalized Korn's inequality and conformal Killing vectors
• Maxwell meets Korn: A New Coercive Inequality for Tensor Fields with Square-Integrable Exterior Derivative
• Nečas-Lions lemma revisited: An $L^p$-version of the generalized Korn inequality for incompatible tensor fields
• Korn's second inequality and geometric rigidity with mixed growth conditions
• $L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative
• Uniqueness of integrable solutions to first order systems with integrable tensor-coefficients and applications to elasticity
• Rough solutions of the Einstein constraint equations with nonconstant mean curvature

Cited by in corpus (5)

• Optimal incompatible Korn-Maxwell-Sobolev inequalities in all dimensions
• Green's functions for the isotropic planar relaxed micromorphic model -- concentrated force and concentrated couple
• Korn-Maxwell-Sobolev inequalities for general incompatibilities
• Matrix representation of a cross product and related curl-based differential operators in all space dimensions
• Korn's inequality in anisotropic Sobolev spaces